This Is the First Volume of the Proceedings of the Committee On

DOCUMENT RESUME SE 004 990 ED 024 576. By- Durst, Lincoln K., Ed. Geometry Conference, Part1: Convexity and Committee on the UndergraduateProgram in Mathematics Applications. Committee on the UndergraduateProgram in Mathematics,Berkeley, Calif. Spons Agency-National ScienceFoundation, Washington, D.C. Report No-16 Pub Date Aug 67 Note- 156p. EDRS Price MF-$0.75 HC-$7.90 Report s, *Curriculum, *Geometr y, Mathematics, Descriptors- *College Mathematics, *Conference Undergraduate Study Identifiers-California, National ScienceFoundation, Santa Barbara volume oftheproceedingsofthe Committee onthe Thisisthefirst Conference, held atSanta Undergraduate Program inMathematics (CUPM) Geometry conference was to considerthe status of Barbara in June, 1967.The purpose of the fended by geometry incolleges at theundergraduate level.The conference, at teachers,involvedlectures on variousaspects of undergraduate mathematics of the relevanceof geometry, analysisof material presented,and an examination In PartIof the proceedings are geometry to theundergraduate curriculum. Sets introduction by WalterPrenowitz and (2) thelectures on Convex contained (1) an of Geometry, by and the CombinatorialTheory of ConvexPolytopes and Applications Branko Grunbaum andVictor Klee. (RP) 6'-E-ao579,9 COMMITTEE ON THUNORGRAHATE PROGRAMIN MAREMATICS

//k

NUMBER 16 AUGUST 1967

CUM%GEOMETRYCONFERENCE

PROCEEDINGS PART I: CONVEXITYAND APPLICATIONS Lectures by BrankoGrUnbaum and VictorKlee

EDUCATION & WELFARE U.S. DEPARTMENTOF HEALTH, OFFICE OF EDUCATION

EXACTLY AS RECEIVEDFROM THE THIS DOCUMENT HASBEEN REPRODUCED ORIGINATING It POINTSOF VIEW OROPINIONS PERSON OR ORGANIZATION OFFICIAL OFFICE OFEDUCATION STATED DO NOTNECESSARILY REPRESENT POSITION OR POLICY.

M ATHEMATICAL ASSOCIATION OF A MERIC A CUPM GEOMETRYCONFERENCE

Santa Barbara,California

June 12 - June30, 1967

PROCEEDINGS OF THECONFERENCE

Edited by Lincoln K.Durst

PART I: CONVEXITY ANDAPPLICATIONS

Lectures by BrankoGrUnbaum and Victor Klee

COMMITTEE ON THEUNDERGRADUATE PROGRAM INMATHEMATICS

"PERMISSION TO REPRODUCE THIS COPYRIGHTED MATERIAL HAS BEENGRANTED By E. A. Cameron Treasurer MAA TO ERIC AND ORGANIZATIONS OPERATING UNDER AGREEMENTS WITH THE U.S.OFFICE OF EDUCATION. FURTHER REPRODUCTIONOUTSIDE THE ERIC SYSTEM REQUIRES PERMISSION OF THE COPYRIGHT OWNER."

Financial support for the Committee on the Undergraduate Program in Mathematics has been provided by the National Science Foundation. A limited number of copies of these Proceedings is available from CUPM, Post Office Box 1024 Berkeley, California 94701. FOREWORD

Conference The Proceedingsof the CUEM Geometry will be issued inthree parts:

(Lectures by I Convexity andApplications GrUnbaum and Klee). (Lectures by II Geometry in OtherSubjects Gleason andSteenrod). ,Groups, and Other IIIGeometric Transformation and others). Topics (Lectures by Coxeter,

The texts printedhere are based onrecordings and were made of the lecturesand the discussions, prepared for publicationby the assistants(Hausner, themselves were able Reay, and Yale). The lecturers the final to make minorchanges and corrections on revision sheets, but an earlydeadline prevented major The typing for or extensivepolishing of the texts. offset was done byMks. K. Blackand the figures were prepared by Mr. DavidM. Youngdahl.

Lincoln K. Durst Executive Director,CUPti CONTENTS

INTRODUCTION

LECTURE TOPICS 3

MEMBERS OF THE CONFERENCE 4

APPLICATIONS OF GEOMETRY 7 Victor Klee

CONVEX SETS AND THE COMBINATORIAL THEORY OF CONVEX POLYTOPES 43 Branko GrUnbaum INTRODUCTION

A Geometry Conference,sponsored by the Committee on theUndergraduate held on Program in Mathematics of theMathematical Association of America, was the Santa Barbara campus of theUniversity of California from June12 to

June 30, 1967.

The conference had its genesis in ameeting of geometers which wascalled

(/' by P. C. Hammer to consider the statusof geometry in our schools andcolleges.

The meeting was held in Chicago on January27, 1966 and was attended by more than twenty mathematicians includingR. D. Anderson, Chairman ofthe Committee on the UndergraduateProgtam in Mathematics, and E. G.Begle, Director of the

School Mthematics Study Group. It was generally agreedthat the immediate

focus of the problem of geometry lay atthe undergraduate level (althoughits

solution was related to questionsinvolving the graduate curriculumand pre-

college mathematics) and that theproblem should be referred to CUPM. In May,

CUPM authorized a smallconference to consider theqiiestion further. A meeting was held in the fall of1966 at UCLA, attended by S. S. Chern, L.K. Durst,

P. C. Hammer, P. J. Kelly, V. L. Klee,Jr., W. Prenowitz, N. E. Steenrod,with

Anderson as chairman.After preliminary discussion of theproblem, Chern pro-

posed a summer conference ofundergraduate mathematics teachers whowould attend

lectures on various aspects of geometry,analyze the material presented, and

examine its relevance to theundergraduate curriculum.Steenrod suggested that

in addition lectures be given onthe geometric underpinning ofother branches

of mathematics. This new and experimental format,which would involve features

of a summer institute and a seminar oncurriculum, quickly received approvalof

the group and of CUPM itself. Anderson appointed a Planning Committeeconsist-

1 of Hammer, Kelly and Klee, with Prenawitz as chairman.

The Planning Committee proposed a conference of four weeks duration but consented to a reduction to three weeks for fiscal and other reasons, despite misgivings that there might not be enough time for a natural development of curriculum discussion.Twenty-one college teachers were chosen as participants by invitation, since the experimental nature of the conference did not require a large membership.

The following mathematicians lectured for a period of one or more weeks:

H. S. M. Coxeter; A. H. Gleason, B. GiUnbaum, V. L. Klee, Jr., N. E. Steenrod.

Shorter series of lectures were given by H. Busemann, G. Culler, P. C. Hammer,

P. J. Kelly and W. Prenawitz. Each lecture was followed by a discussion period--

this helped to contribute a freshness of spirit to the discussion since questions, remarks and challenges did not have a chance to be forgotten or lose their cutting edge. Several discussions on curriculum were scheduled as the need arose.The program was supplemented by the showing of several films produced by the College Geometry Project of the University of Minnesota.

An important innovation was the selection of three assistants to the lecturers, younger mathematicians who were responsible for writing up the lecture notes and leading discussions on the material and its relation to the undergraduate curriculum.M. Hausner, J. R. Reay, and P. B. Yale were chosen for these assignments and carried them out with singular dedication.

The text which follows gives a record, sometimes in summary form, of the lectures and discussions of the conference.

Walter Prenawitz

2 LECTURE TOPICS

Herbert Busemann

The Simultaneous Approximation ofn Real Numbers by Rationals. An Application of Integral Geometry to theCalculus of Variations.

H. S. M. Coxeter Transformation Groups from the Geometric Viewpoint.

Glen J. Culler Some Computational Illustrations ofGeometrical Properties in Functional Iteration.

Andrew M. Gleason Geometry in Other Subjects.

Branko GrUnbaum Convex Sets and the Combinatorial Theoryof Convex Polytopes,

Preston C. Hammer Generalizations in Geometry.

Paul J. Kelly The Nature and Importance of Elementary Geometry in aModern Education.

Victor Klee Applications of Geometry

Walter Prenawitz Joining and Extending as Geometric Operations: A Coordinate-Free Approach to n-space.

Norman E. Steenrod Geometry in Other Subjects.

3 MEMBERS OF THE CONFERENCE

Russell V. Benson Murray S. Klamkin California State College,Fullerton Ford Scientific Laboratory (Lecturer) Gavin Bjork Victor L. Klee, Jr. Portland State College University of Weshington John W. Blattner Rev. John E. Koehler San Fernando Valley StateCo/lege Seattle University Herbert Busemann (Lecturer) Sister/4. Justin Merkham University of SouthernCalifornia St. Joseph College Jack G. Ceder (Visitor) Michael H. Millar University of California, Stanford University Santa Barbara H. Stewart Mbredock Sacramento State College G. D. Chakerian University of California,Davis Richard B. Paine H. S. M. Coxeter (Lecturer) Colorado College University of Toronto Walter Prenawitz (Chairman) Glen J. Culler (Lecturer) Brooklyn College University of California, John R. Reay (Assistant) Santa Barbara Western Washington State College Andrew M. Gleason (Lecturer) Paul T. Rygg Harvard University Western Weshington State College Neil R. Gray George T. Sallee Western Washington StateCollege University of California, Davis Helmut Groemer James14. Sloss (Visitor) University of Arizona University of California, Bradko GrUnbaum (Lecturer) Santa Barbara University of Washington Norman E. Steenrod (Lecturer) Preston C. Hammer (Lecturer) Princeton University Pennsylvania State University George Stratopoulos Melvin Hausner (Assistant) Webtr State College New York University Robert/4. Vogt Norman W. Johnson San Jose State College Michigan State University William B. Woolf Mervin L. Reedy University of Washington Purdue University Paul B. Yele (Assistant) Paul J. Kelly(Lecturer) Pomona College Untversity of California, Santa Barbara Raytond B. Killgrove California State College, Los Angeles PROCEEDINGS APPLICATIONS OF GEOMETRY Lectures by Victor Klee (Lecture notes by Melvin Hausnerand John Reay)

Lecture I.

During the past few years, wehave witnessed the effects ofthe many

efforts to revise and improvethe mathematical curriculum(SMSG, CUFM, etc.).

These efforts have proved tobe widely influential andbeneficial--much more However, the possi- so than we mighthave imagined when they wereinitiated.

bility of further progress should notbe ignored. In particular, an informal

session at the 1966 Chicago A.M.S.meeting made it clear that manygeometers

favor a broad further revisionof the geometry curriculum fromthe kindergarten clearly through the senior undergraduatelevel. Since such a broad attack was

not feasible, it wasdecided to start at the collegelevel. Hence we are

gathered here.

Let us consider thefollowing four questions.

decline? 1. Is collegiate geometryinstruction in a state of

2. If so, is this undesirable?

situation? 3. If this is undesirable, whatshould be done to improve the

4. What should a college geometry courseconsist of?

Our assumption is that questions1 and 2 have affirmative answersand

that questions 3 and 4 are onlyslightly different.Once question 4 has been

answered, the next step will be towrite the appropriate books.

that of (It is profitable to comparethe situation of geometry with books by algebra. In algebra, thousands ofstudents have learned from the Hence the van der Waerden orfrom Birkhoff and MacLane or theirimitators.

algebraists have a common experiencewhich serves to strengthentheir subject.

g /7 I do not believe theuniversality of these books was aresult of general agreement on what belonged in analgebra course at the time they werewritten.

Rather, it occurred because thebooks were written so well thattheir readers

came to think thatthis was the only natural way toproceed.)

What are the desirablecharacteristics of an undergraduatecollegiate geom- Except etry course? In my opinion, they shouZdinclude the following points.

for the first and third,these apply as well to anycollege mathematics course. if a) The course should be n-dimensional,and even infinite-dimensional letdown for this is possible with little extracost. Anything less will be a

the student and will minimizethe usefulness of the course. For example, b) There should be a unifyingtheme throughout the course. differentiable manifolds or it might be thestudy of certain objects, such as such as invariance or symmetry. convex bodies.Or it might be some notion, their structure c) It is more important tostudy the geometric objects, One and their properties, than tohave an estheticallypleasing axiom scheme. Thus should use the mlst powerfulapproach rather than the mostesthetic one. in nature only if no the axiomatic basis shouldbe specifically "geometrical"

loss of efficiencyresults from this. other areas of mathe- d) It should emphasize thepoints of contact with

matics. technology wherever e) It should includeapplications to science and

possible. mentioned in order to whet thestudents' f) Unsolved problems should be

interest.

Subjects which mightsatisfy these criteriafor a geometry course are n-dimensional projective geometry, differentiable manifolds,algebraic topology,

8 other possibil- and the geometry of convexbodies.There are of course several ities.

My candidate is a course on convexbodies. As far as point a) is concerned, the subject is easilycarried out inn dimensions. As GrUnbaum pointed out, many of the methodsused for n-space areessentially two- or three- dimensional, making motivation andintuition very easy inn dimensions even theme is the study of the for the inexperienced.As to point b), the unifying

properties of convex sets. This is very pictorial,and close in spirit to the

familiar Euclidean geometry, butit makes much more contactwith modern mathe- c) is to use matics. The method which seems tofit the criterion of point

real vector spaces as the settingfor the theory. Hawever, if Prenowitz's

"join geometry" approach canbe developed in moredetail, it might very well GrUnbaum's be used instead. The possibilities for pointf) are excellent (cf. of convexity with other areas lectures). Finally there are many connections Let us of mathematics, and manyapplications in science andtechnology.

mention just a few. tools are separa- 1. In functional analysis,three of the most important

tion theorems for convex sets(the basic separation theorembeing equivalent for convex sets, and fixed- to the Hahn-Banachtheorem),, extreme-point theorems

point theorems for convex sets. Inclusion of these topics in aconvexity geometrical course servessimultaneously to teach thestudent some interesting

facts and to prepare him for alater course in functionalanalysis. be brought 2. In a very natural way, anumber of topological notions can characteristic can be into a course on convexity.For example, the Euler

approached in a combinatorial waybased on convex sets. Having defined subdivisions of simplicial subdivisions andproved that a simplex admits

9 combinatorial lemmas(taking arbitrarily small mesh,then by using Ky Fan's (perhaps in two hours) suchbasic perhaps one hour), it ispossible to prove Borsuk-Ulam antipodal mapping results as Brouwer's fixedpoint theorem, the Lebesgue tiling theorem, etc. theorem, the invariance ofdomain theorem, the obtaining further propertiesof One can then proceedto use these results in convex sets. combinatorial properties of convexsets, various 3. In a study of the of course there are many combinatorial identities enterin a natural way and do not seem to arise contacts with graphtheory. At present, matroids probably naturally in such a course,but techniques nawbeing developed will (Jon Folkman has a newcombinatorial change this situation in afew years. related to linear notion which is relatedto positivebases as matroids are

bases.) best viewed from thepoint of 4. In inequalitytheory, many results are of convexity Lnsummability view of convexfunctions. There are applications

theory and in manyother parts of mathematics. mathematics which useconvexity There are many areasof modern applied

belaw. in a significant way. Some of them are listed theory, the study of convexpolytopes forms the 5. In linear programming

essential geometricbackground. programming theory, convex orconcave functions 6. In much of nonlinear

form the background. well as Liapounoff's 7. Control theory usesthe separation theorems, as measure. theorem asserting theconvexityof the range of a vector

classification uses separatinghyperplanes in 8. The theory of pattern Schlafli's theorem on the numberof regions a fundamental way,and in particular

10 determined by a finite set of hyperplanesin general position through the origin.

9. Parts of information theory areclosely related to packing problems for

convex bodies.

I readily confess that I have notattained a sufficiently high degree of

organization to cover all the above topics in a one-yearcourse, but I believe

it to be possible. The unifying theme makes this less of ahodge-podge than

it appears to be.

Discussion.

The question of a unifying theme wasbrought up. In particular, Klee

admitted that this was not so apparent inBirkhoff and McLane, although the

notion of a group permeates this text. In this connection Klee feltthat a

course based on the notion oftransformation as a unifying theme would be too

bland in nature, with too few deep or excitinggeometrical results.

Point c) of the ideal geometry course wasdiscussed in some detail. This

led to the general question of whetherthe course should be content-oriented

or foundation-oriented,with Klee and most of the vocal participantschoosing

the former.However, when it was suggested that amathematics course without

axioms is simply not a mathematics course,Klee stressed that he did not wish

to eliminate the axioms--hewanted to use an efficient axioms systemand not

concentrate on esthetics. In any event, the underlyingaxioms would be stated.

When the question of the importanceof n-dimensional space wasbrought up,

it was pointed out that a junior orsenior level course was underdiscussion

11 and that this should not cause problems at thatlevel.

Finally, there was general agreement that ayear's course on convex bodies should not be the sole upper level geometry course. It should be one of several. But Klee felt that one geometry course whichhas some depth is better than the customory hodge-podge course.

12 Lecture II.

We shall naw discuss someapplications of convexity whichmight be in-

cluded in the sort of courseGrUnbaum is discussing. These are not applica- of tions in the strictest senseof the term. Rather, they illustrate the sort objects which come from IIpure appliedmathematics" in which one takes certain

applied mathematics and then onestudies them for their awnsake, just as one has would any other mathematicalobject. (Note that much of mathematics

developed in exactly this way.)

We now introduce a fewmathematical notions which formthe necessary

backgrouad for the material. These will be elaborated inGrUnbaum's lectures.

C Definition. An extreme point p of a convex set C is a point p of for set-theoretic such that C (p) is convex. (We use the notation A ^dB the convex setC if p difference.) Equivalently, p is an extreme point of points of C. is in C but is not the mid-pointof any segment joining two polygon are the vertices For example, the extremepoints of a plane convex

of that polygon. constitute one of the Extreme points and techniquesfor manipulating them

three aspects of convex setswhich are most important inconnection with (rhe other two applications of convexity toother portions of mathematics. The importance of are separationtheorems and fixed-pointproperties.)

extreme points isdue in part to the followingfact. thenK is Theorem. If K is compact, convex,and finite dimensional,

the convex hull of its extremepoints; that is, K = con ext K.

(This result goes back toMinkowski. In the correspondinginfinite-

dimensional result of Kreinand Milman it is necessaryto take the closed

convex hull of the extremepoints.)

13 If we abandon the restriction of compactness, we have the following analogous result.

Theorem. If Kis closed, convex, finite-dimensional, and line-free

(no line is contained in K), then Kis the convex hull of its extreme points and its extreme rays; that isK = con(ext K U rex K). (An extreme ray of a convex set Kis a half line contained in K which is not crossed by any segment in K.)

A ray which is not extreme An extreme ray

Amother important reason for studying extreme points is in their application to a wide variety of practical optimization problems. For example, suppose one has a cost function which is to beminimized. In many cases, this function f is defined.over a convex set C and iu many applications f is affine, or at least concave. A basic property of such functions is as follows.

Theorem. If a concave function attains its minimum on a line-free closed convex set, it does so at an extreme point.

For the above result, as for the remainder of these lectures, we are assuming that the convex set is finite-dimensional.

We would like to indicate a more or less constructivemethod of obtain-

ing this minimizing extreme point, starting from an arbitraryminimizing

point. Let us consider for the sake of simplicity a compact set,and

let us first consider the simpler problem of finding an extreme point.

14 GrUnbaum will probably describe a quick inductiveproof of the existence of

extreme points, but the followingprocedure is more constructive in nature.

In the diagram, we start at any point xl. If it is not extreme, it is

the mid-point of some segment. Extend this segment in one direction as far as

possible, to a boundary point x2. We continue the process withx2, and in this way we obtain a sequence x1,x2,... of points. We claim that this procedure must terminate after.at most d+1 steps, where d is the dimension of the set.

To see this, note that the point x3 cannot be on the line x1:x2, by the

special choice of x We claim similarly thatx cannot be in the plane 2. 4

xxx and so on. The reason is indicated in the following flgus. If it 123

were possible to continuealong the line L in the plane x1x2x3, then, since

we are in a convex set, theshaded triangle would be in that set. This implies

thatxx can be extended beyond x which is a contradiction of the choice 12 2 of x Similar arguments hold for higher dimensions.Of course, the pictor- 2' ial argument can be made formal. Now, if f is a given concave function,

assumed continuous for the purposes of illustration,then a similar process may

be applied to find its minimum.For if the minimum is attained at xl, and if

x is not extreme, it is an immediate consequenceof the notion of concavity 1

15 that the minimum is also attained at one or the other endpoint ofany segment

throughx , say atx We may then continue as above to prove the stated 1 2' theorem.

The required definitions of concavity and convexity, along with the pic-

torial idea are as follows.

Definition.Let C be a convex set, and let f be a real-valued function on C. Then

1) f is concave iff(Xx + (l-X)y) Xf(x) + (1-X)f(y) for all xand y inC and all X between 0 and 1:

2) f is convex if the above inequality is reversed. Equivalently, f is convex if -f is concave:

3) f is affine if it is both convex and concave; the above inequality is replaced by an equality:

These formulations translate the usual geometric notions. Thus, f is concave, convex, or affine if the graph lies respectively above, below, or on any chord.

It is clear why the minimum of a concave function on a line segment must be attained at an endpoint.For if not, we would have a graph with points:

which is clearly not possible for a concave function.

We now consider a specific class of minimization problems, namely, the transportation problems. Such a problem (an m X ntransportation problem) is specified by a positive m-vector a mi ai > 0, and a positive n-vector b (b1,...sbn), bi >0, where 41.1ai =13.1 bi. We suppose that we have a homogeneous, infinitely divisible commodity located at the sources

16 All,aileTheamounta.of the commodity is at the sourceAi.The problem is to transport this commodity tothe destinations (or sinks) Bj in such a way that the total amount bj of the quantity arrives at A solution of

matrix x = ( xij ), wherex is the amount of the this problem is an m x n ij material from source A which is sent to the sink Bj Thus, the solution i xis a matrix which satisfies theconditions:

a 1. x 0 xl1x12 1 ij

2, e.;.1xij ai (row sum) am bn 3. Em x = b (column sum) bl i=1 ij j

The rectangle is a convenient pictorialdevice to help us remember the size of xand the row and column conditions. The solutions, as a set of m X n matrices, form a subset ofErna. Furthermore, the defining equations and inequalities forxshow that the set of such x's is a convex polytope.To see this, note that each ofthe mninequalities x 0 defines a closed ij halfspace, and each of the equalitiesdetermines a hyperplane which is the

intersection of the two closed halfspacesdetermined by that hyperplane. Thus

the set of matricesx satisfying the required conditions is anintersection

of finitely many closed halfspaces, namely apolyhedron. It is clearly bounded,

since each component is bounded byEi ai. Thus it is a polytope.

Definition. If a andbare vectors as above, the setof matrices x

satisfying the above conditions is calledthe transportation polytope T(a,b).

Now suppose there ate costifunctions cij, where cij(a) is the cost of

sending an amount afrom source Alto sinkB Then the cost of a

particular transportation scheme xis simply

C(x) E cij(xi )

17 In most practical problems the functions c.. will be concave, whence C is 13 also concave. When the c 's are linear this is an example of a problem in ij linear programming, namely to minimize a linear function over a convex polyhedron.We know that the minimum is attained at some vertex of the polytope.

(A vertex is an extreme point of a polyhedron.) It is also easy to see that the set of minimizing points is the convex hull of the set of minimizing vertices. The subject of linear programming involves various algorithms to find these minimal vertices. We should like to point out that sometimes this is not-a simple problem even though there are only finitely many vertices. The number of vertices may be very large, so that a direct search is impractical.

Further, the polyhedron is generally not given in terms of its vertf.ces but only as the solution set of a system of linear inequalities,

It is not the purpose of these lectures to discuss practical methods of solving linear programming problems.However we can sketch the practical technique usually employed. First find one vertex. (Often this is not trivial.) There are computationally practical ways whereby, having a vertex of the polyhedron, one can "look" at all of the adjacent vertices.

Of the adjacent vertices, choose the one which gives the smallest value to the cost function or which in some other sense represents a maximum improvement. Then continue the process. It must terminate at a minhmiz- ing vertex, as follows from a simple theorem in the geometry of polytopes.

This is all for linear cost functions. For concave functions the situation is much more complicated.

Although linear programming problems are concerned with finding an optimizing vertex of the feasible region, there are closely related problems in which one may want to know all of the vertices. An example would be a

18 transportation problem in which an enemy is choosing the non-negative coefficients c.. of the linear cost function (subject to E c.. = I) in the 13 13 hope of maximizingC(x).

We now study the geometry of T(a,b).We precede this sequence of results by the remark that the subject itself is not of overwhelmingmathematical interest. It is given simply as a sample of what could bedone, in a course treating convex polytopes, to increase the students' contactwith the more applied aspects of the subject.

Theorem 1. dim T(a,b) = (m-1)(n-1)

We prove this result in several steps.

a) The equality constraints (the row and column conditions)define a flat of dimension (n-1)(n-1). Each of the row and column sum conditions defines a hyperplane.The claim is that the intersection of these hyperplaneshas the required dimension.Momentarily, the inequality constraints are being ignored.

To prove a), we note that there arem+nlinear functions here. Let, for th .th example, p (x) = E. x. (= i-- row sum) andy (x) = E. x. (= 3 column ij i J l sum). These m+nlinear functions are not linearly independent, since

E. p. = E. y., However, if we delete y , the remaining m + n - 1 linear 1 1 J J n functionals are linearly independent, (For example, we may easily arrange to make any one of these equal to 1, while the others equal 0.) Thus the rank

of this system of linear functionsequals m + n - 1and the dimension of the

flat in question ismn - (m+n-1) =(m-1)(n-1).

%mark. Using the theorem ofCaratheodory, this result shows that each

feasible transportation scheme (point of T(a,b)) is a convex combination of

(m-1)(n-1) + 1 or fewer vertices. For n X n doubly stochastic matrices,

the vertices may be identified as the permutationmatrices. Thus, everS7

19 2 n X ndoubly stochastic matrix is a convex combination of n -2n -I- 2 or fewer permutation matrices.)

b) For each i and j, there exists a vertexx of T(a,b) such thatx.. > O. 13 With no losslaigenerality, we may assume i = m, and j = n. Then we use the "southeast corner rule,"x = min(a ,b) and proceed at each stage to mn mn fill in the southeast corner

a sb m n

a m-1

. . a 0 . 0 0 m a b 1 n with as large an entry as possible; that is

xij = min(a. - E. x. 2 b.- E. xrj). 3

Then we claim that the matrix so constructed is a vertex. The proof is by an easy induction which is left to the reader.

c) There is a point ofT(a,b) which is interior to the positive

or thant determined by the inequalities2c. > 0. 13

To see this, let p be themn vertices determined in partb). Now

take their average, p = (E. . p..)/mn. Then p is a convex combination of 1,3 13 vertices, hence an element of the polytope. But p clearly has all of its

entries positive and hence is in the interior of the positive orthant.

Finally, the proof of Theorem 1 follows immediatelyfrom a) and c), since

T(a,b) is the intersection of a flat of dimension (m-1)(n-1) which passes

through an interior point of the positive orthant.

20 Discussion.

The discussion clarified the final partof the proof of Theorem 1. In addition, the formal definitions of convex and concavefunctions were presented, as gtven in the notes. The importance of these functions were stressed by Klee, insofar as practical maximum and minimumproblems were concerned. For example, the minimum of a concave function isalways attained at an extreme point, while for convex functions, a localminimum is a global minimum.

Problems of minimizing or maximizing nonconvexand nonconcave functions tend to be much more difficult.

Klee pointed out that transportation problems are areal application, not a contrived one. In particular, a significant portionof present day computer time is devoted to the solution of linearprogramming problems.

The question of computing volumes of polytopes wasbrought up as a practical one in geometric probability. The difficulty even for dimensions

2 and 3 was indicated. Two-dimensional problems can be done with some in- genuity, and higher dimensional problems with alarge amount of symmetry can also be handled. But otherwise, nobody can do it.Klamkin pointed out that

POlya, in his thesis, considered the problem of finding the volume cutoff by two parallel hyperplanes in a cube. Explicit formulas were given and were derived by Laplace transforms.

21 Lecture III.

We continue thediscussion of the propertiesof transportation polytopes (A facet of a d-dimen- with a theorem about thenumber of facets ofT(a,b). the sional polytope is a faceof dimension d-1. In the study of polytopes, vertices) are usually faces of extrem dimension(namely the facets and the great importance as the most interestingfaces.) The theorem is not of any will learn things far as applications areconcerned, but in proving it we about the structure of theproblem which are of interestfor the applications.

Assume from now on that m g n.This is really not anessential limitation,

since mathematically there is nodistinction between sourcesand sinks.

and two facets Theorem 2. T(a,b) has exactly one facet when m =1 and mn, when m = n = 2.The number of facets isotherwise between (m-1)n

each possibility beingrealized by some transportationpolytope. We prove a number ofstatemedts, Proof. The first two cases aretrivial.

each of which is easy byitself.

I a particular a) Each facet of T(a,b) is of the form (x E T(a,b) each set of this form is a coordinate x = 03. (This does not assert that ij the number of facets.) facet. Note that this does givethe upper bound on orthant, but this T(a,b) is the intersectionof a flat with the positive that when orthant itself has exactly mnfacets. Thus it suffices to prove P n Vthen F is Pis a polyhedron,Vis a flat, and Fis a facet of

the intersection ofV with som facet ofP.

(For example if P is a cube, Va plane

1.0 NUN albowlell11111111111111...110.11 cutting its middle, thefacets ofP n V 01111111VO

are the fouredges of the (middle)square.) Pnv t

The proof uses acharacterization of facets

Fof a polyhedron asmaximal convex subsets

22 of the boundaryof the polyhedron. and only if itsdimension b) A set of the formin a) is a facet if is (n-1)(n-1) - 1. form is the intersectionof the This is evident, sinceeach set of this hyperplane. polytope (of dimension(n-1)(n-1)) with a supporting does this form give afacet Thus we want todecide for which pairs (i0j)

of the polytope.

(i,j). Note that a is the Let a =Em a =En h , and fix c) r=1 r c=1 c ai + bj is the sum ofthe total amount of thecommodity shipped,while received by the sink B. amount shippedfrom source Aiand the amount

03 misses T(a,b) a < a. + b iff (x E T(a,b) I xij = T(a,b), iff (xlx =03 determines a vertex of a = ai +bj ij T(a,b). iff (xlx =03 determines a facet of a> ai +bj ij orthant and the flatdetermined by This gives a relationbetween the positive intersection of asupporting hyperplanewith a the set of a). In general the dimension. polytope may be a facet or avertex, or aface of some intermediate

cannot occur. But here theintermediate possibilities picture of thesituation.Each Rather than provethis, we consider a Assume (for notationalsimplicity, point of T(a,b) is a matrixx, as shown.

and without lossof generality) that a 1 (i,j) = (msn). The question is:What can

we say aboutmatrices x inT(a,b) with

x = 0? If Et + b> a, i.e., mn n bn > a - a= a there can be no m i=1 is b non-negative xin T(a,b) for which 1

b= x Ltn?-1a If x = 0, for any such xwould give i=1 in i=1 mn n

23 be am + b= ait follows that all elementsin the last row and column must n as large as possible.Thus the rest of the matrix mustbe zero, i.e., the this point is a vertex of hyperplane meetsT(a,b) at exactly one point, so and column may the polytope. Ifam + bn < a, the entries in the last row be chosen less than the maximum amount (e.g., xln < al, etc.), and aresub-

Thus ject only to the two restrictions el x = b and E 111.=1 x = am. i=1 in n mi (within restrictions) we have a"frce choice" of (m-2)(n-2) elements of the

and column, reduced matrix and (n-2) and (n-2) free choices in the last row Thus i.e., (im-2)(n-2) + (m-2) + (n-2) =(m-l)(n-1) - 1 free choices.

=03 determines a facet ofT(a,b). [xixmn Note that this result isof interest because if an opponentis choosing

if the cost functioncij, he may make use of thefollowing fact:

scheme must send somethingfrom a + b >.a then clearly every feasible i

source A to sink Bj.

d) If aNa+ b and aga +b with r4 u,s0v, then r s u v in distinct rows and m = n 2. That is, if we have two critical points

columns of x, then m = n = 2.

a - au = E , a a , This is clear since a a-b=E b by r r s n = 2 with the second and fourth eWebeing "=" only if respectively

and m = 2.Thus41 these "bad points" (i,j) must be in the same row or

in the same column,and there can be at most nof them (since m n). where But these points (i,j) are precisely theplaces in thein X n matrix

c). Thus at least [x E T(a,b) 1 x = 0) is not a facet, by part ij This proves the mn n = (m-1)n ofthese sets must be facets ofT(a,b).

and inn theorem except for showingthat all possibilities between (31-1)n

are actuallyrealized.

24 We now turn to the number of vertices ofT(a,b). We first characterize the vertices in a useful way. This characterization will not be explicit enough to allow us to count them in general.Leading up to this, I would like to mention a standard correspondencebetween matrices and bipartite graphs which will be useful in some of the counting processes.

Associated with any m x n matrix we have anundirected graph whose nodes or vertices are divided into two sets; one setof nodes corresponding to the rows of the matrix, theother set of nodes corresponding to thecolumns th th of the natrix.We connect the I: row node to the column node if and only if the corresponding entry in the matrix is non-zero. For example, this matrix and graph correspond.

(rows) 1 6 0 0 0 0

0 3 5 8 0 0

0 0 9 2 7 0

0 0 0 4 411100014 (columns)

(numbers in each node are raw or column sums)

These graphs are associated with the transportationproblem in a natural

way, since we can thinkof the raw nodes as sources and thecolumn nodes as

sinks. The arcs may then be labeled with the amountof shipment along that

arc.We will prove that the feasibletransportation schemes (i.e., points

x E T(a,b))which are vertices of T(a,b) are exactly thosewhose graphs

are trees or"forests" (unions of a finite set of disjointtrees).

Definition. A 122p for the matrix xis a sequence (i1d1)(i1d2)

1) where we successively change only oneof the two numbers

25 i or j, such that

a) there are no repetitions, and

b) the matrix has non-zero entries in all the indicated places.

Equivalently, a matrix has a loop if and only if its graph has a circuit, since the only way you can leave a "row" node is to go to a "column" node in a bipartite graph, and to leave a column node you must go to a raw node. Thus the following theorem has a geometric as well as graph-theoretic interest.

Theorem 3.The vertices of the polytope T(a,b) are exactly those x E T(a,b) such that the matrix x admits no loop.

Proof.Supposex E T(a,b) and xhas a loop (11,j1)(i2,j1)...(is,j5)

listed) > 0. Define a matrix y by (i j 5) Set 2e = min (x I (i,j) 1, ij = xij if (i,j) is not listed in the loop,

= x e if (i,j) is in the loop ij where e is alternately added and subtracted as we go around the loop. This has the effect of leaving all row sums and all column SUMS fixed and hence y,i(x+y), and i(x-y) are all members of the convex polytopeT(a,b). But x =i(x+y) + i(x-y) and thus xis not an extreme point.

Conversely, if x E T(a,b) ev ext T(a,b) we will show that the matrixx has a loop. If x is not a vertex of T(a,b) then there is some y such

thatx y E T(a,b), and the row SUMS and column sums ofy must necessarily all be zero. Now choosey 0 0. Since all row and column sums of y are ili1 zero we can continue to choose pairs (i,j) which define a loop in yfor which the correspondingyij are non-zero.Sincex+y and x-y, being in

T(a,b) must have all non-negative entries, it followsthat each xij must be posittve and me thus have a loop in matrixx. This completes the proof.

26 following are equivalent: Theorem 4. For any matrix, the

a) x admits no loop

b) x E T(a,b) is a vertex of T(a,b)

(obtained by deleting certain rowsand columns) c) every submatrixof x A row or column is admits a distinguished row orcolumn. (Definition: distinguished if it has at most onenon-zero element.)

r c I distinguished rows d) anyr x c submatrix has at least whenr g c(distinguished columns when c r). distinguished row or adistinguished e) every squaresubmatrix has a column distinguished row and adistinguish- f) every squaresubmatrix has both a ed column has at most 2k - 1 non-zero g) every squaresubmatrix of order k entries. be put into gtaph The conditions of thistheorem could, of course,

that the bipartitegraph of the matrix theoretic terms. For example, c) implies is distinguished x must have atleast one "dead-end," To say a certain row node is joined to at most oneother corresponds to sayingthat a particular row

if we pick any set ofnodes containing the node.Condition e) asserts that nodes, and the arcswhich join these chosen same numberof row nodes as column dead-end in that subgraph. Condition nodes, then there mustbe at least one node and a dead-endcolumn node in f) asserts thatthere is a dead-end row

each such subgraph. the vertices of areally None of theseconditions allows us te count they do allow us tolook at complicated transportationpolytope, although that they are notvertices. certain pointsx E T(a,b) and say immediately

27 Discussion.

Housner pointed out that there are only finitely many combinatorial types of the polytopeT(r,b) c: en and yet the combinatorial structure depends upon

,m+n the continuous parameter (a,b) E L He asked what determined the boundary

n of a set of points (a,b) E Elm+ which gave a transportation polytope

T(a,b) c en of a particular combinatorial type. (Definition: Polytopes

P and P' are of the same combinatorial type if the lattices 5(P) and

5(P') of their faces are isomorphic.) Klee answered that this was typical of a class of similar questions that could be asked, most of which arehard.

Perhaps you could prove that two transportation polytopes inen had the same combinatorial type provided a certain characteristic functionx(a,b) assumed the same value on the pairs (a,b) and (a',b') which defined the given transportation polytopes.The characteristic function x would be defined in terms of the class of all subsets of the set of m + ncoordinates of (a,b).

Certain game theory results have applications to econamics or business, but no one suggested any peaceful applications of the game-theoretic transportation problem in which an enemy is determining the cost function. Klee remarked that since all "bad points" of a matrix x (i.e., pairs (i,j) for which a.< a+ b in the notation of Theorem 2) must occur on one row or one i column, there will be one critical source or sink in these cases whose lines of communication must be closely guarded.

28 T.ecture IV. AO.% .4:1/101/11111

We will continue our discussion of transportation polytopes, but will omit the details of some of the counting arguments. These would be similar in spirit to those done above, but c.onsiderably more complicated and time-consum- ing.

We first consider the possible number of vertices. A transportation problem determined by (a,b)c:Em-1-11 is called degenerate if the sum of the elements of same proper subset of(ai Ilgig m3 is equal to the sum of some proper subset of (bi 1 g j g n3. A problem is degenerate if and only if some feasible solution has a graph which is disconnected.

Theorem 5. For a general m x ntransportation problem (in g n), the minimum number of vertices is

(n-m41): and this is achieved in certain examples.For m x nnon-degenerate problems -1 the minimum number of vertices is WI This gives a complete solution for the minimum number of vertices.

Only partial results are known about the maximum number of vertices of the general m x n problem.

Theorem 6. For m x ntransportation problems, the number of vertices of T(a,b) is achieved by non-degenerate problems.

Sketch of proof. Each T(a,b) is the intersection of a posittve orthant with a flat.This polytope can clearly be approximated as closely as desired by a polytope which represents a non-degenerate problem. One then makes use of the fact that the number of vertices of a polytope is a lower semi-continuous function of the polytope.

29 We next consider themaximum number of verticesfor regular

(= non-degenerate) problems.

Lemma 7. Two vertices ofT(a,b) which have the same pattern of non- zero entries mustbe identical. characterized as matrices withoutloops. Proof. Verticesx and y were i(x+y). Clearly If x and y have the same non-zero pattern,then so must must also be a vertex. this cannot happen unlessx = ybecausei(x+y)

Since every extreme pointof T(a,b) has a graph which is a tree, we can of get upper estimates ofthe number of verticesby counting certain classes

trees. number of vertices Theorem 8. For any m X ntransportation problem the n-1m-1 is at most the number m n of spanning trees ofthe complete bipartite

graph with m vertices in oneset, n in the other. rather than forests (i.e., We can restrictourselves to (connected) trees because we have restrictedthe unions of disjointtrees) in the above theorem, n-1m-1 problem to the non-degenerate case. It should be noted that m n is the in the following number of "generalized" verticesof the d-polytope T(a,b) hyperplanes determined sense. Each genuine vertex is theintersection of d hyperplanes may or may not be in by d-facets. However the intersectionof d the the polytope and thus itmight not be a vertex. The above number counts bound on the number of number of such intersectionsand thus is only an upper n-1m-1vertices can you get genuine vertices. One could ask, howclose tom n be a very hard with a transportation polytope?At present this seems to

problem, although some progresshas been made. supply l/n, and As an example, suppose wehave n sources, each with

Then several observations maybe made n+1 sinks, each with demand 1/(n+1).

30 about the graph of a vertexof this polytope:

least two arcs leading tosinks, since a) Each source must have at l/n > 1/(n+1). exactly two arcs leading tosinks, since any b) Each source must have edges. Thus the bi- (connected) tree with 2n+1 nodes must have exactly2n partite graph mightlook like this:

graph which is 2-valent ateach source node, c) For every bipartite be assigned to the graph. there is a feasibletransportation scheme which may

For the graph shownabove the assignedscheme will be

k/n(n+1) of the total (where the amountkshown on each arc denotesthat

commodity movesaiong that arc). if there were n d) The above argumentswould work in a similar way

each source was (01)-valent. sources, pn + 1sinks, and therefore the number of verticesto the transporta- e) The problem of determining conditions is again a matterof tion polytope when weimpose these symmetry )n-1 (n!) when there aren counting trees. The number turns out tobe (n+1

equal source nodesand n+1equal sink nodes, or

(111n4.1)n."1 (on)! (119n

31 when there are n equal sources and pm + 1 equal sinks.

Our discussion of transportation polytopes willend with the following comment. Much of the material presentedabove is well-known but some of it is new.A much more careful exposition, accompanied by completereferences and proofs, can be found in a forthcoming paper by V. Kleeand C. Witzgall.

Discussion.

It was pointed out that the upperbound in Theorem 8 was frequently quite

a bit too large. Several other problems were mentioned which appearsimilar

to the transportation problem, but whichapparently are related only in that

combinatorial techniques are needed.

The symmetry of the last transportation problemexample (nequally

strong sources, n+1 equally strong sinks) leads to a question aboutthe

symmetry of the corresponding polytope T(a,b) and the determination of its

facial structure.Klee mentioned that GrUnbaum would consider theproblem of

determining the number of facets of a d-polytopewith v vertices, and

remarked that it would be interesting tohave similar results for the class

of polytopes which had a group of symmetriesof a particular order.GrUnbaum

remarked that almost no results of this type areknown. Klee pointed out that

theorems on the facial structure of polytopes areof direct interest to linear

programming problems.

Woolf asked how proponents of a geometry coursebased on convexity, as

GrUnbaum and Klee have discussed, would meet theclaim that geometry courses

based on algebraic topology anddifferential geometry would be of more use to

32 that to do something a student inhis further work. Klee was of the opinion and level of maturity is well in those two areas, amuch better background In particular many necessary than todo a careful treatmentof convexity. elementary. If a topics in convexity are moreintuitively accessible and background student were going on toresearch in those areas,then a thorough but for an undergraduate at the undergraduatelevel could be preferable, in Klee's opinion,be general training, a coursebased on convexity would, conference, it was felt better. Paul Kelly pointed outthat in planning this neglected was in differential that the place college geometrywas least lecturers who would concentrate geometry, and that itwould be better to invite the college level. on the other partsof geometry which arein more trouble at

33 Lecture V.

We will next consideranother topic related to linear programming.

Perhaps this should be called aproblem whose answer would lead to anapplica-

tion rather than an application of geometry.Much of this lecture can be

found in the last chapters ofGrUnbaum's book, Convex Polytopes.

Linear programming problems arefrequently solved by choosing a vertexof

the polytope described by thelinear constraints, investigatingthe neighboring

vertices, and thus searchingfrom one vertex to the next forthe maximizing

vertex. In such a search process we movealong the edges from one vertex to

the next, and it would be useful tohave results which give upperbounds to

the number of such iterations that a computermight have to perform. (The

usefulness of a computer in solvingproblems often depends upon whether you

know the problem will take at most2 minutes, 2 days, or 2years!)

Definition.Let Gbe a graph (for example, the setof edges and

vertices of a polytope). The distance

8(x,y) between verticesxand y is

the length of the shortest path(counting

edges) between x and y. The diameter

of ofGis max(6(x,y) I x,y vertices G3.

If P is a polytope, 6(P) denotes the

diameter of the graph of P. For example, if P is a cube, then6(P) = 3.

n Definition. Ab(d,n) is the maximum diameter of ad-polytope with is omitted.it is the samedefini- facets. (rhe b is for bounded, if the b

tion for the class of (possiblyunbounded) polyhedra.)

Definition. A graph G is d-connected if it cannotbe separated between

edges. any two verticesby removing fewer than d vertices and their adjoint

34 (See GrUnbaum's For example, thegraph of the cube shawnabove is 3-connected.

lectures, Section 12, forfurther related results.)

Before stating the maintheorems, we make a fewobservations. Equivalently, for Lemma 1. The graph of a d-polytopeis d-connected.

d disjoint paths each pair of distinctvertices of a d-polytope,there exist

connecting these verticeswith only the end-pointsin common.

The proof is omitted. of a d-polytope with Lemma 2. (GrUnbaum-Motzkin) The maximum diameter

vertices is

r:EZZ1+ 1. d d independent paths Proof. Suppose xand y are vertices. Consider

shortest. Internal to each from x and yand let kbe the length of the

2, so path there must be atleastk-1vertices. Thus v (k-1)d v-2 k-1 andk g [(v-2)/d] 1. Lemma 3. Ab(d) is achieved by asimple d-polytope with n facets.

(P is simple if each vertexis d-valent.)

Proof sketch. A small amountof "wiggling" of eachface of the polytope d-valent, and will not increase thenumber of faces, but canmake each vertex and edges out of oldmulti-valent in general willproduce additional vertices

vertices.ThusP may be assumed tobe simple.

This is useful in the3-dimensional case.

Lemma 4. If d = 3, Ab(3,n) =411]- 1. edges, and faces, Proof. Lettingv, el fdenote the number of vertices,

is simple impliesthat 2e = 3v. Thus e f = 2by Euler's formula. P Ab(3,f) g [(v-2)/3] + 1 =[2f/3] - 1. Easy m 2f -4. By Lemma 2 and 3,

examples shaw the bound isactually assumed.

35 or whenn g d+5, Theorem.Ab(d2n) =[151511113-d + 2 when d 3 except thatAb(4,9) = 5. Ab(d,n) The proof is omitted. These give the only specificvalues of that are known. outstanding problem in linear The determination ofAb(d,2d) was a long programming, and the conjecturethatP(d,2d) = dwas "proved"for d 5 4 Thus problems of this before it was recently shown tobe false even in E .

type are difficult eventhough they are veryeasily stated.

The conjecture of Hirsch wasthat A(d,n) g n - d. This has been shawn

4 of the conjecture false even in E . However the bounded version conjecture that is very easy tostate, (d,n) g n-d is still open. A related b polytope can be joined by apath and yet unsolved is: Any two vertices of a of which does not "revisit" anyfacet (i.e., considering a path as a sequence

vertices, a path may travelaround the vertices of afacet as far as desired,

but once it leaves a facet it may neverrevisit the facet later inthe

true, then the boundedversion of sequence). If the latter conjecture were form of the Hirsch's conjecture wouldbe true. Actually, the following stronger

conjecture might be true. In any cell-camplex

any two vertices maybe connected by

a path not revisiting anycell. It is

not known whetherthis conjecture is

true even for2-dimensional cell complexes

3 that these embedded in E . I would expect

conjectures are probablyfalse when d gets large, say d 9. be found in Details of these and similarresults and problems may

Grunbaum's book.

36 Applications of Helly'sTheorem. theorem or related There are several applicationsto science of Helly's could well be brought into a results (see GrUnbaum'slectures, Section 5) which do with molecular classroom discussion onHelly's theorem. One of these has to

genetics.

In studying DNA moleculeswhich specify the structureof certain phage The phage particles particles, the following investigationprocedure was used. Each mutant arises exist both in certain standardforms and in mutant forms. DNA from some blemish in the geneticstructure. If the structure of the

molecule is represented by agraph, we are interested inknowing the structure graph which contains the of this graph, and particularlyin the part of the

blemished part which producesthe mutant forms. There seemed to be experi-

mental reasons for assumingthat the blemished portionof each graph was a Further there was sub-arc of the graph rather thanjust some random subset. The research- a way of tellingwhen two of the blemishedsub-arcs overlapped. pattern is consistentwith ers then askedwhether this observed intersection linear part of the the hypothesis that allblemished intervals came from a

graph of theDNA molecule. Or is it necessary to use morecomplicated

structures than a line toobserve the intersectionpattern?

This leads to the followingmathematical problem.Determine completely of intervals on a what intersection patterns canarise from finite families

limitation at once; namely,if line. Helly's theorem on the line gives one is such that each twohave a common a finite familyof intervals on the line apply directly point then their intersectionis non-empty. But this does not The intersection here for we areconcerned only with pairwiseintersections. matrix of zeros and ones,with pattern itself could bethought of as a square

= 0 for the mutants i and j, and 1 0 in the (i,j)-place when Ii n I

37 this in the (i,j)-place if n I. 0 0. The question then becomes, is matrix consistent with theassumption that the intervals arechosen from a line?

A more convenient way ofdescribing this is in graph-theoreticlanguage.

The intersection graph of anyfinite family of sets is thegraph whose nodes represent the sets, and twonodes are connected by an edgeif and only if the corresponding sets have non-empty intersection.For example, the intersection graph of the line segments on theline at the left is the graph atthe right.

3 16 : rrn.

We will say that a graph is aninterval graph if it is the intersectiongraph

of a finite number of intervals onthe line. The problem is now to character-

ize all interval graphs in auseful way.We will state only one of the more

satisfying characterizations ofinterval graphs.Mbst parts of the proof of

this theorem would be understandableand interesting to undergraduates. A

graph has the rigid circuit propertyif each circuit is decomposable into

triangles, that is, each circuitwith more than 3 edges has crossings.

Theorem. G is an interval graph if andonly if

(1) it has the rigid circuit property,and

(2) each three vertices of the graphadmit an ordering so that every

path from vertex 1 to vertex 3either goes through vertex 2 or isonly one

edge away from vertex 2.

To finish the story, it wasshown that for the experimentalresults which

were obtained,the intersection graph of theseveral hundred blemished

38 Examples intervals in the DNA molecules didindeed form an interval graph. useful and interesting things of this sort point out tostudents that there are properties of convex sets on a that can be said evenabout the intersection line.

References of the S. Benzer, On thetopology of the geneticfine structure, Proceedings National Academy of Sciences U. S. A. 45(1959) 1607-1620. 206(1962) 70-84. The fine structure ofthe gene, ScientificAmorican for complementation mappings E. A. Carlson, Limitationsof geometrical models of allelic series, Nature191(1961) 788-790. matrices with the consecutive l's D. R. Fulkenson and O.A. Gross, Incidence 70(1964) 681-684. property, Bulletinof the American MathematicalSociet

P. C. Gilmore and A. J.Hoffman, A characterizationof comparability graphs 16(1964) 539-548. and of interval graphs,Canadian Journal of Mathematics of a finite graph by a C. G. Lekkerkerkerand J. Ch. Bolawl, Representation 51(1962) set of intervals onthe real line, FundamentaMathematicae 45-64.

An application ofbarycentric coordinates.

This application of geomotryis of interest inconnection with mineral situation engineering. It deals with thephase diagram of a multicomponent solutions are formed. which is isobaric, isothermal,and in which no solid example with just three For the sake ofillustration, let us consider an interested in all the primary components,labeled A, B, C. Suppose we are from combiningdifferent possible chemical compoundswhich could be formed stable at the amounts of these threeprimary componentsand which would be

39 given pressure and temperature. Each such compoundD can be represented as

a point in thetriangleABC, where

the unique barycentriccoordinates

of D with respect toA, B, and C

determine the relative amounts of

components A, B, andC that are

present in D. (Generally it is

possible only in parts of inorganic

chemistry to determine a compoundin

terms of the amounts ofthe elements

of primary components fromwhich it

is formed. This cannot be done with

carbohydrates, for example, butthere are large areas withininorganic chem- this tri- istry where this ruledoes apply.) Along with all the vertices in has a angle which represent thedifferent possible stable compounds, one

decomposition of the triangle intosmaller triangles, as shown.Of course, decompositions possible, but for a given set of verticesthere are many such

in fact there is only onesuch decompositionwhich represents the following

important aspect of thesituation. Suppose we started with aparticular of each mixture of the compoundsA, B, andC, and the relative amounts the obvious way. determine the barycentriccoordinates of the point K in that all sorts of decom- If this mixture isheated to a high temperature, so and position take place, andthen is slowly cooled tothe original temperature 3 compounds pressure, theresulting mixture does notconsist of the original will consist of the com- A, B, C.Rather almost all ofthe resulting mixture AED which pounds which arerepresented by the verticesof the small triangle

40 of the compounds A, E, and contains K. Furthermore, the relative amounts

't in terms ofA, E, D will bedetermined by the barycentriccoordinates of and D.

It is easy to seethe importance of this incertain refinery processes. this type of scheme isuseful in Given a particular typeof ore, for example, and in what amounts.This determining what kinds ofcompounds can be produced

and c) and determining amounts to knowing thecompounds in the ore (A, B, E) lies in, and what itsbary- which small simplex(triangle) the ore (point in the 2-dimensional case centric coordinates are. This is rather trivial In fact, this is since you can just draw apicture and see what isgoing on. coordinates just what people do. In the higherdimensional cases, barycentric is the must be computed inthe standard ways.The problem mathematically, d-dimensional simplex following. Let po,p1,..,pd bethe vertices of a and which contains the smallersimplex whose vertices are cloAy...cid,

know whether suppose z Econv(po,p12...2pd). We wish to the affine coordinatesof z z Econv(cj02q1,...qd), and if so, what are

in terms of the qi. there really are situations This sort of problemshows the student that coordinates is desirableand for which the actual computationof barycentric

computing barycentriccoordinates a naturalthing to do.The first ideas in determinants, but there are are justthe usual techniqueswith quotients of lead to the attempt to practical refinements ofthese techniques which soon

compute the coordinatesin the most efficientmanner.

41 Discussion.

It was asked how thedecomposition of the triangle intosmaller triangles was determined fromknowing just the 3 originalvertices and the locations of knew from the experiment the interior vertices. Suppose for example, that you that DEFG was a part of thediagram, and you wondered which of the two diagonals DForGEshould be added. The standard technique is

to form a mixturewhose relative

amounts determine thecoordinates

of the point K, heat it, then

cool it, and observe whethercompounds

E andGor compounds Dand F

result.Analogous techniques existfor the higher dimensional cases.

References for this whole topic are:

P. A. Beck, Journal ofApplied Physics 16(1945) 808-815. Chemistry 52(1948) 698-760. L. A. Dahl, journalof Physical and Colloid

V. Klee, Problem inBarycentric Coordinates, Journalof Apslied physics Vol. 36, No. 6(1965)1854-1856.

P. M. Pepper, Journalof Applied Physics 20(1949) 754-760.

Johnson pointed out thatanother application ofbarycentric coordinates legislature seats among has been in the partitioningof an integer number of certain percentage of various political parties,each of which has received a the vertices the total vote. In this setting thepolitical parties determine which represent certain of a simplex, and thesimplex is subdivided into areas this is in Mathematical divisions of the legislature.An article describing

Smapshots by Steinhaus.

42 CONVEX SETS AND THE COMBINATORIALTHEORY OF CONVEX POLYTOPES Lectures by BrankoGrUnbaum (Lecture notes by John Reay)

Convexity is a subject that can easilybe taught to undergraduates at the junior or senior levels, and there areseveral good reasons for doing so.

First, it is possible to reach significantresults without the necessity of a strong background in other fields. There is no prerequisite of any extensive techniques. Secondly, convexity is a tool which may beapplied in many other fields of mathematics, including number theory,functional analysis, complex variables, and others.Also it develops a student's geometric intuitionand intuitive comprehension of the proofs andtheorems. It introduces combinatorial reasoning which has application in many otherfields. One of its main advan-

tages, which is rare in other areas of geometry,is that it is possible to

introduce students to many open problems early inthe course. This makes it

possible to show that there is much work yet to bedone, and problems that have

challenged our best efforts to find a solution.This type of course can be

quite a contrast to many courses whichwould lead the student to believe that

all mathematics is done and comes in a completedpackage. Finally, many proofs

of convexity have arguments in 2- and 3-dimensional seJcewhich makes for easy

understanding, even though the proofs are valid in higherdimensions.

These notes cover only a small fraction of thedirections in which con-

vexity has developed. In the first part of the notes(Sections 1-9) a basic

introduction is given to convexity and someof its main tools, applications,

and references. Sample proofs are included, but manyof the results are left

in the form of unproved statements orexercises. These sections are not meant

to be a complete review of thetopics usually covered in a course oncouvexity,

43 foundtion for the combinatorialtheory but they are rather meantto supply a of convexity. This of convex polytopes, oneof the rapidly expanding areas designed to give an second part of the notes(Sections 10-14) is primarily problems and basictechniques intuitive grasp ofthe important results, open

of this area. TABLE OF CONTENTS

Part I. 45 1. Algebraic prerequisites properties of convex sets 51 2. Definition and elementary 60 3. Separation and support 72 4. Convex hulls 78 5. Helly's theorem 85 6. Extreme and exposedpoints; faces and poonems 93 7. Unbounded convex sets 100 8. Polyhedral sets 103 9. Bibliographic notes

Part II. 109 10. Number of facets ofpolyhedra 128 11. Related results andproblems 3 137 12. Polytopes inE 149 13. Polytopes with rationalvertices 152 14. Bibliographic notes

44 1. Algebraic prerequisites. easily proved, will be The following facts,generally known from algebra and used--without special mention--inthe sequel.

reels R, and if A = (a2... ,a) is a If Xis a vector space over the 1 n

sequence of n vectors in X, any expression ofthe typeEi=1 Xiai,where We shall say that eachX. E R, is a linear combination ofthe vectors in A. = Xn = 0; Ais linearly independentprovided 4.1 Xiai= 0 implies Xi =

ife.lx.a.=Oispossiblewithsomex.00, we shall say that A is 1=1 if and only if some member linearly dependent. Hence Ais linearly dependent

of Ais a linear combinationof the other members ofA.

A maximal linearly independent sequence iscalled a linear basis of X;

this clearly means that each vectorx E X is alinear combination of the

elements of a linear basis ofX.Moreover, this linear combinationis unique;

basis. its coefficients are thecoordinates of x relative to the given linear

Any two bases of the same space Xhave the same cardinality, known asthe

d, (linear) dimension ofX. If Xis d-dimensional for somefinite cardinal

of X, ane ifA = (al,...,ad) and B = (hl,...,bd) are two linear bases

there exists a regular lineartransformation Tof X onto itself such that and Ta. = b for i = 1, Conversely, if Ais a linear basis of X i andT is a regular linear transformationof X onto itself, then in X (Tal,...,Tad) is a basis forX. Any linearly independent sequence

may be extended to alinear basis for X.

If X is d-dimensional, a sequenceA = (a1,...,an) of vectors in X is

linearly independent if andonly if the matrix

45 a . . . a nl nd

a arethecoordinatesofa.relative to a has rank n, where 2eee,aid given linear basis ofX. d The linear dimension of the Euclideand-space E is d. A convenient

d linear basis ofE is the standard basis formedby the vectors el = e2 = ed = (0,...,0,l).

d, and ifx If X and X*are two vector spacesof dimension 1 d form a basis ofX, and form a basis of X*,a mapping Tfrom 12* 2 d d * X.x. . This mappingT is toX* may be defined byT(E1. X.x ) = E. X 1 an algebraic isomorphismbetween X andX*; moreover, if both Xand X* are topological vector spaces,Tis a homeomorphism. d The scalar product (a,b) of vectors a,b E E is the real number defined by

(a,b) =4=1alpi .

The most important properties ofthe scalar product are

(a,b) = (b,a)

(Xa,b) = x(a2b)

(a..03,c) = (a,c) + (b,c)

(a,a) 0 with equality if and only ifa = 0.

If (a,b) = 0, then a andb are said to be orthogonal, toeach other.

If (a,a) = 1, then a is called a unit vector. In the sequel, the letter

u (with or without subscripts) shall beused only for unit vectors.

46 A hyperplane H inE is a set which maybe defined as

y 4 0, anda. An open H = (x EEd (x,y) = al, for suitable y EEd, I

I (x,y) > a) (respec- halfspace (closed halfspace)is defined as (x EEd

for suitable y EEd,y 0, anda. Clearly, tively (x EEd I (x,y) 2:oi)

d is also an openhalfspace for y 0, similarly for (x E E 1 (x,y)

d x of X and "orthamormalize"it by X and E , take any basisx I"d putting y and

74k-1(xkai) Yi for k = 2,...,d. 4k Li=i

el, ,e.ti form a linear Letei = yi /7577.,y1.) for i = 1,...,d. Then d basis of X and, ifelp...pedis the standard basisofE , the trans-

cl formation T such thatTe= e' T(E Xe) =Ed X et is an isometry i t=1 ii i=1 i i

of the required type.)

The affine facts listedbelow may either bedertved fram their linear

counterparts, or be provedindependently.

the reals, and if A =(a1...pan) is a If Xis a vector space over 1

of the type 4.1 X.a , where sequence of elementsofX, any expression i of the elements of A. all X E R ande X = 1 is an affine combination i=1 is an We shall say that A is affinely independentprovided no member of A

47 affine combination of theother members of A; otherwise Ais 2ffinely

dependent. in R2 A is affinely dependentifand only if there existX12...,Xn

0, such that E4 X a= 0and e X = 0. not all equal to i=1 ii 1 and only if the sequence A = (a ,a) is affinely dependent if 1' n

(a - a2...2a - a) is linearly dependent. 1 n n1- n d, and if the elements a of A If X is of (linear)dimension i X, then have coordinates a ...2a relative to some linear basis of '112 id A is affinely independentif and only if the matrix

cll cYld 1 .

. a a 1 nl nd

has rank n. X. Any maximal) affinelyindependent sequence in X is an affiae basis of affine basis ofX. If Ais a linear basis ofX then A U (0) is an d x2y E E is The set of all affinecombinations of two differentpoints and the line14(x2y) = ((1-X)x XyI X real). If x',y' E L(x,y)

xl 0 y', thenL(x'2374) = L(x2Y). E H, If a set H has the propertythatL(x2y) c: H whenever x2y

x 0 y, we call H aflat, or an affine variety. Clearly, the set of all is a affine combinations formedfrom all finite svibsetsof a given set A A.The flat; it is denoted byaff A and is called theaffine hull of d d all family of all flats in E contains E , 0, all one-pointed sets,

if all HIs are flats, lines, all hyperplanes; also,it is intercectional: a

aff Aof a set A may equivalentlybe so isn H . The affine hull a a

48 A. The formation of defined as the intersectionof all flats which contain aff(x+A) = x + aff A. (Note the affine hull istranslation invariant, i.e.,

general.) thataff(A+B) = aff A + aff B is not true in d VofE Every flat His a translate H = x 4.V of some subspace of a certain dimension and is therefore isomorphicto the Euclidean space

is thenr = dim H =dim V. r the dimension of H (and of V) affinely independent points,but Each r-dimensional flatcontains r+1 each (r+2)-membered set of its pointsis affinely depeuSent. d preserves A transformation Tof E intoEnis affine provided it affine combinations; that is,

T(4=1aixi) =4=1aiTxi

a = 1. xi EEd and aIs are realssatisfying whenever i 1=1 i again Images, or inverse images, offlats under affinetransformations are d 1 from E to E= R is flats. In particular, anaffine transformation A of any point determined by Ax =(a,x). If a 0 0, the inverse image by A d a E Ris a hms21.121 inE , denoted by

pce) = CxEEd Ax = ,x) = ce)

this form for suitable Conversely, each (d-1)-flat inE" is expressible in

aand a and is,consequently, a hyperplane. the corresponding linear ones The main advantageof the affine notions over Hence, among otheradvantages, their is their invarianceunder translations. origin" at some advantageouspoint. use in proofsavoids "choosing the the reals is irrelevant The assumption thatXis a vector space over field of characteristic to many of thedefinitions and factslisted above. Any

49 0 could be used instead of thereal numbers.

Discussion.

It was generallyfelt that the affine notions(affine independence, trans-

formations, etc.) were usuallyunfamiliar to the students. It was suggested

that this could be due to alack of affine topics in mostlinear algebra

to give an adequatefounda- courses. Opinions about how much time was necessary Klee suggested tion in affine topicsvaried from one or two hoursupwards. affine, positive, letting the word "blank"stand for any of the words linear, "blank or convex,and then develop the generalideas of "blank independence," See hulls," etc. as far aspossible with comparisons oftheir differences.

Klee [1].

50 2. Definition and elementary propertiesof convex sets.

Let X be any vector(linear) space over the reals, offinite or infinite dimension. A set K C Xis called convex provided

(*) whenever x,y E K, then all pointsof the (straight-line)

segment determined by xand y belong toK.

Denoting by R the fieldof real numbers, condition(*) may be reformulated

as g 1, (**) whenever x,y E K andA E R satisfies0 g

then Ax + (1-70y E K.

Examples of convex setsin any real vector space X: whole space X: (i) the empty set0; any single point; the generally, any flat (ii) any (linear)subspace of X and, more

(i.e., translate of asubspace) in X. (i.e., any set of the (iii) any (closed oropen) halfspace of X a E R and yis a type (x E X y(x) al or (x E X y(x) > a), where yis real-valued, non-trivial distributivefunctional on X; thismeans that whenever 51(x) 0 0 for somex EX, andy(ax + PY) = ay(x) + pp(y)

x,y E X, a, E R). closed or open ball in X (iv) if X is a normed space, any

or (x E X where (i.e., set of the type (x E X IIix-x011 cv) Ilx-x011

xo E X,aE R, anda >0). from Convex sets aregeneral enough to appear inwidely different fields, they have elementary geometry tofunctional analysis. On the other hand,

sufficiently much structure toallow significant results.

51 In the sequel we shallmostly assume that X is thed-dimensional

d require this restriction, Euclidean space E , though many of the results do not

or may bemodified to hold in more general spaces. d We list naw a fewfundamental properties of convex setsinE; additional

properties will be discussedlater. d 1. The family of all convex sets in E is intersectional; that d means that for anynon-empty family (Kv) of convex sets in E the inter- d secLion n K is of the same type.All closed convex subsetsof E also v v all com- form an intersectional family,and so do all bounded convex sets, or

pact convex sets.

2. Any convex combination of pointsbelonging to a convex set

K CEdbelongs toK. This means

i = 1, (:**) 1, al,...,a E R2 ai 0 for whenevern n

and En a= 1, then x xn E K implies i=1 i 1"."

E. cex. E K. 1=1 i Clearly, condition (**) used in defining convexity is thespecial case as is easily seen n = 2 of (***). On the other hand, (**) implies (***),

an 1, and use by induction on n: assume without lossof generality that

the identity ai _17 xi, Li=l CY ii Cnxn

where X =En-1a d d' 3. If A is an affine transformationof E intoE , and if

AK = tAx x E K) is a convex K cEd andK' cEd' are convex sets, the I 1K' E 10). set, and so is the inverseimage by A ofK', A = EEd

52 Hence, in particular,the convexity of a set Kis not affected by any This may translation (x+K). homothecy (aK), or orthogonaltransformation. in which a suit- be used in proofs to reducethe more general situation to one

able point is at the origin. (See, for example, the proof ofTheorem 6.10.) d and K are convex sets in E , then 4. If K 2

K+ Km (x+ x2I xl E x E K) is a convex set. 1 2 1 12 2 2 Note that we shall usethe notation -K = (-1)K = [-x1 x E

"subtraction" of sets (K K) is not and K K= K + (-IC). Hence 2 1 2 1 2 1 the inverse operltion of the(vector, or Minkowski) "addition"of sets

CK+ K). 1 2 d el K and 5. If Kc:E is convex, then itstopological closure

and its iaterior int K are also convex.Thus ifx,y E int K

0 N X g 1, then (U{ 4. (1-70y) E int K. d Denoting the distancebetween pointsx,y E E by Hx-yll =

the convexity ofcl K follows from the observationthat if

Ilxi-xil <8 and Oi-51 < 8 then, for0 g X g 1, (1"X)fiyi-Yll <6. Analternate (Xxi + a-7051) (Xx + (1-X)Y)11

proof of the convexity of cl K follows from 1, 3, and4 above and the centered at observation that el K = n (K+ 0),where Bis the unit ball fact that there is some the origin. The convexity of int K follaws from the

open d-ball Bcentered at the origin for which x+Bc K and y + BC K.

Thus if 0 g. X g 1 and ifu E B then

(Xx + (1-X)y) + u = X(x+u) -I.(1-X) (y+u) E K

and therefore (Xx + (1-X)y) + B C K, so int K is convex. d is convex, 6. Similarly, ifx E int Kand y E K whereKC E

and if0 < X 1, then Xx + (1-X)Y E int K.

53 As in the last proof, ifx + B c:K thenit follows that

(Xx (1-X)y) + X13 C K.

Remarks d E may be 1. The intersectionality ofthe family of all convex sets in For example, we used to define convex sets, orcertain types of convex sets. subsets of shall see in Section 3that the family of allclosed convex proper d intersectional family ofsubsets of E may becharacterized as the smallest d Similarly, the family of allcompact E which contains all closedhalfepaces. d subsets of convex subsets of E is the smallestintersectional family of d d of E may be E containing all closed d-balls. All convex proper subsets

obtained analogously by startingwith"semi-spaces". (Semi-spaces may be

explicitly defined in various ways: a particularlyappealing definition--

though unsuitable if oneintends to use semi-spaces todefine convex sets-- does not contain a given is: a semi-space is anymaximal, convex set which

point.)

Other instances of thisapproach to, convexity will bediscussed 1ater.

(***) are frequently 2. Modifications of conditions (**) or Euclidean spaces, encountered in the definitionsof various families of sets in

or in moregeneral vector spaces.A few examples are: Whenever Type of set Together with x andy the set also contains cY2P E R and

ax + py no othercondition (i) (linear) subspace cy + 0 = 1 (ii) flat (affine variety) ax + py a >0 (iii) cone olx

a ... 0,13 > 0 (iv) convex cone ax + py

(v) convex set ax + (3y

(i), (ii), and Linear, affine andpositive combinations (generalizing combinations were (iv)) are defined similarly tothe fashion in which convex combinations defined by (***). See Klee [1]. Some of the results on convex combinations (or for we shall seelater have analogues forthe other types of

some of them). been studied; such as, A number of other notionsrelated to the above have y the set contains for example, the mid-pointconvexit (where with x and combinations with arbitrarilyprescribed or i(x+y)). Some results are known on B. GrUnbaum, restricted sets of coefficients. See the article by L. Danzer, and Motzkin [1]. and V. Klee [1]--referred to hereafter as DGK[1]-- transformations of 3. It is sometimes convenientto consider mojective d d called convex sets inE A transformation Pfrom E to itself is

projective provided it ispossible to express P in the form

Ax + b = _ (C220-1-45

d b,c EEd, 8 E R, where Ais a linear transformationof E into itself,

and at least one of c and 8 is different from 0.

The transformation P is not defined onthe set

N(P) = Cy EEd I (c,y)+6 = 01.

55 in which caseP is an affine The set N(P) may be empty(if c = 00 8), is a hyperplane(which, if A is transformation. If c 0 then N(P) of the r(2_21essize regular, is mapped by P onto the"hyperplane at infinity" d d-space containing E). d transformation as above, P If K is a set inE andPa projective provided K n N(P) =0. Generalizing is said to bepermissible for K

property 3 we have: d transformation If Kc:E is a convex setand ifP is a projective

permissible forK, then PK is convex. has been modified in numerousways to make it 4. The notion of convexity over thereals. In suitable for settingsdifferent from vector spaces We shall mentionhere a few particular see DGK[1], Section 9especially. instructive to investigatethe results we shall of these variants;it is rather their proofs to make encounter with respectto themodifications needed in

them valid for thedifferent "convexities." field, convexity in anyvector space overQ (i) IfQis any ordered This notion isuseful in clarifying may bedefined by the analogueof (**). compactness assumptionsin various the role ofcompleteness, closure, or is that in whichfinite- theorems. A particularlyinteresting special case numbers are considered. dimensional spaces overthe field of rational

that the most usefuldefinition of convexityin (ii) Experi/2nce has shown d the d-dimensionalprojective spaceP is: d (d-1)-dimensional A subset Kc:P is convexprovided there exists a d straight line subspace of H of P such that K n H =0, and if for each

d connected. Lof P the set K n L is -.tither empty or

56 if we are dealing with asingle Note that thissetting is not interesting d transformation carries thehyperplaneH convex set in P since a projective the set K onto a setequivalent to a set onto the hyperplaneat infinity and

d 1 does not hold forprojective convexity. inE Note that property have been found useful. (iii) A number of notionsof spherical convexity definitions and furtherreferences.) (See p. 157 of DGK[1] for four different d later is the following. Let S The one we shall haveoccasion to mention d d+1 d S =(x E E ) = I). A denote the unit d-spherein E , that is,

d provided the set coneK CEd+1 is convex, set Kc: S will be called convex 0 (half-lines) originating at where coneK denotes the unionof all closed rays 0 point ofK. the origin0 E E and passing through a

Exercises assertions made above. 1. Prove in detail the d-ball 2. Prove the convaxityof the (closed) unit d d,,, d B=(x E E 04 1) c: E .

3. Prove the convexityof every d-simplex. d midpoint convex subsetof E is convex. 4. Shaw that each closed, d in E . Shaw that if every 5. Let tKv) be a family of convex sets intersection, then denumerable subfamily of(K) has a non-empty

n K 0 0. v v d "equivalent" provided thereexist x EEd 6. Let K!, K"c E be called Prove that thisis an equivalence and A > 0 such that K" = x AK'. equivalence classes of convex relation, and thatthere are eleven distinct of the distinctclasses of convex subsets ofE = R.What is the cardinal 2 subsets of E?

57 K = -K), d 0 as centerof symmetry (i.e., 7. Let E:c:E have the origin Show that thefunction and assume that Kis compact andthat int K 0 0.

cp defined forx EEd by

tp(x) = inffa OixEaKj

d is a norm on E . (That is x = 0; for all x EEd,with cp(x) = 0 if and only if (i) (10(x) 0 for allx,y EEd (ii) cp(x+y) cp(x) + yo(y) x EEdand all a E R.) (iii) cp(ax) =lalcp(x) for all is a metric on Ed. Equivalently, showthat p(x,y) = cp(x-y) d subset ofE and Bis the (topological) 8. If K is a compact convex E), then K = C. The boundary of K, andifC = [ix +iy I x,y E "compact" may not beweakened example of a closedhalfspace shows that

to ''closed."

Discussion.

various definitionsof spherical Johnson pointed outthat one of the d and raised thequestion of convexity isequivalent to t)nvexity in P , finite fields. This appears to defining convexity invector spaces over

be a difficultproblem. of a slightly strongerversion of 6: Klee suggestedthe following proof and K is convex,then u = Xq +(l-X)p E int K, Ifq E el K, pE int K,

for0 < 1.

58 the origin of radius e for As before, assume B is an open ball at

B centered atq of radius which p BC K. Consider the open ball

which is the reflectionthrough the point u. Then X . e, that is, the ball l-A lies in the qEciK impliesthat there is somexEK(110. Thenu xand some y E p B. relative interior ofthe line segmentdetermined by

The proof thenproceeds as in 6.

59 3. Separation and 2222E1. d Let K denote a convex subset of E , d ?.40. We shall first prove:

1. int K 0 0if and only if K contains d+1 affinely independent

points.

In other words, the interior of Kis non-empty if and only if the affine d hull aff K of Kis the whole space E .

Indeed, if a maximal affinely independent setof points ofKcontains d

or fewer points, theiraffine hull contains K but has dimensiond-1 or

less--hence int K = 0. On the other hand, if K containsaffinely independent

points xo,...,xd, then K contains the open simplex

0 CE Xx Ed X. = 1, X > 0,...,Xd > 0) i=0 i I i=0 0

henceint K 0. d For Ac: E let rel int Aand rel bd A denote, respectively,the

interior and the boundary of A considered as a subset of the flat aff A k (Which is isometric to someE). From 1 there follows

2. rel int K 0wheneverK 0 0.

Indeed, the following stronger resultholds:

3. int Kc: rel int K = rel int(c1 cl(rel int g) = cl K.

The proof of 3uses 2 (or 1) and the technique used in the proof of 2.5

(i.e., result 5 in Section 2 ).

d A2are separatedk a Let A1,A2c:E ; we shall say that AI and

Ed (u,x) = X) (where u EEd is a unit vector, hutrplane 1.1(1,X) = (x E I

and X E R) provided Alis contained in one and A2 in the other of the

=Ht(udo= (x EEd u,x) X) and If = 117(u,X) = closed halaaa921 I

d 1 (x E E (u,x) g X).Note that a set may beseparated from itself, for 2 example, a line segment inE We shall say that non-empty setsAiand A2

60 provided Aiis contained in oneand A2 are strictly,separated, by H(u,X) 71- and int H. in the other ofthe aaa halfspaces int H

It is immediatethat AI andA2, then H separates cl A 4. If H = H(u,X) separates 1 int Al and int A2. and cl A2, and itstrictly separates separation is The fundamentallemma on strict d sets inE such that 5. If K and K are compact convex 1 hyperplane which strictlyseparatesK1 and K2. K1 n K2 = 0, there exists a

y2 EK2) = 8(K1,K2). Since Proof. Let a = inf(11y2-y111 I 571 E Kl, and disjoint, a > 0 and there existx E K K and K are compact 1 1 1 denotes the hyperplane -x = a. Then, if H and xE K such that 1 2 2 2 1

x-x)) the convexity of )), and if H= H(x-x (x H(x-x , (x1 ,x-x 2 1 22 221 2 1 2 1 2 - (= int H implies that KCHc: int H andK c:H i K 1 1 2 2

H 2

the hyperplane Hence, for any X such that0

61 then z of K were in the interiorofH-, Indeed, if some point I by convexity, clearly some point on theline segment [z,x1]would be inK1 and be closer to x2than xl. Note that this argumantis 2-dimensional, even d though the proof is valid in E Also note that this is theonly point where was to obtain the convexity hypothesis enters.The only use made of compactness

the existence ofxl andx2.

Regarding the separation wehave d such that 6. If K and K are bounded convex setsin E 1 d separated by a hyperplane if aff (K1 U K2) = E , then K1and K2 may be

and only if

red int K n rel int K= 0. 1 2

shall prove here only the The "only if" assertionis an easy exercise; Tim generality (see 3 and 4) "if" part of the theorem,assuming without loss of

that K andK are closed (hencecompact). For 0 < e < 1, let 1 2 (l-e)(-y0 + K2), where Kl(e) = xo (1-e)(-x0 Ki) andK2(e) = yo

xE rel int K andyE rel int K are fixed. Then 0 1 0

rel int K. = U K (e) for i = 1,2, 1 0 < e < 1

K,(e') c:rel int K (e) for0 < e< e' < 1 and i = 1,2.

are compact anddisjoint, 5 tmplies the existence Since K(c) and 1(2(e) Ki(e) and of a hyperplaneH(e) = H(u(e), X(e))which strictly separates

H(e) intersecto the segment [x02370], the set 1(2(e). Since each is bounded. Together with the compactnessof the unit (X(e) I 0 < e < 1) d-1 this implies the sphere S which contains the set (u(e) 1 0 < e < 1),

existence of a sequence (01,...,en,...), llm en m 0, for which the following

62 hyperplane u = limu(en).Then the limits exist X = lim X(e) and

rel int Kl and rel int K2, hence also H(u,X) is easily seento separate

K and K2. 1 based upon thefollowing assertion: An alternateproof of 5 can be given d d iS proper subsetofE and C= E f**0 C IfC is a non-empty convex 2 I halfspace andbd CI = bd C2 = a also convex, thencl C. is a (closed) Assume K and K are disjoint convexsets hyperplane. (See Hammer [1].) 1 2 d following Lemma: Either d ofE . We use the I. n E and xis any point where cony A or elseK2 nconv(tx) U Ki) = 0, K1 nconv([x) U K2) =0, contains A. (See Section 4.) denotes the smallest convexset which Lemma add x to one Well-order all pointsin the space,and with this 1 which disjoint convex setsK K one of of the two setsK to obtain 11, 21' i these two c:K In a similarfashion expand containsx and for which K x K K one ofwhich contains 2* sets to two newdisjoint convex sets 122 22'

two disjoint convexsets Kt and 11 By transfiniteinduction, we obtain

d By the firstassertion these sets whose union isE and for which Kail!. Thus it which forms their commonboundary. are separatedby the hyperplane that for some point x, suffices to establishthe lemma. Its denial asserts be conv((x) U K1) 0. Thus there must n conv((x) UK2) it 0 and K2 n

Ed and points pointski E Kiand a point x E

yi E Ki nrel int cony

Y2 E X2n rel int conv(xlk1).

63 E K1

the existence of a point This two-dimensionalsituation clearly leads to z E K1 n K2, a contradiction. Thus 5holds.

hypotheses, but the Both of 5 and 6 may bestrengthened by weakening the In 52 basic ideas of theseproofs already appear in theabove formulations. Kibe compact. for example, we maydemand that only one of theclosed sets andX(0) Another application of thecompactness argument(relative to u(e) allows one to strike "bounded"from 6and to establish d such that aff (KU K2) =Ed, If K and K are convex sets in E 1 7. 1 2 only if then K andK may be separatedby some hyperplane if and 1 rel int Kn rel int K= 0. 1 As simple consequencesof the above we have d is the intersectionof all the 8. Each closed convex set K in E d Each open closed (or of all theopen) halfspaces of E which contain K. d d ofE convex set K inE is the intersectionof all the open halfspaces

which containK. d C is a convex set suchthat 9. If K is a convex set inE and if and C. bd K, then there exists ahyperplane which separates K

d and let X E R. The Let A (7- E , let x EEdbe a non-zero vector, hyperplane of A providedeither hyperplane H(x,X) is called a supporting

64 in the first case we or X = inf((x,a) 1 a E Al; X = sup((x,a) I a E A) hyperplane of A indirectionx shall say thatH(x,X) is the supporting denote it by H(A;x). (or with outward normalx), and we shall

From 9we have: (in particular, Ed is convex, andif cc bd K is convex le. If K c bd K), there exists asupporting hyperplane H if C is a single pointof

of K such that C c: H. can also begiven whenC is the The followingindependent proof of 10 with center p and let ybe a single point p. Let B be a closed ball K.Nowy cl K since p must point inB which isfarthest from the set

which is closest toy. Let H be inbd K, so there is apointz E cl K As in previous arguments, be the hyperplanenormal to [z,y] through z. then H must bethe desired yand K are separatedby H, so if z = p, then the hyperplane supporting hyperplaneand we are done. If z p interior of B. Thus some point parallel toH and through ymust meet the (See Botts DJ.) u E B mustbe farther from Kthan y, a contradiction.

A partial converseof 10 is d then each supportinghyperplane 11. If K(= E is compact and convex, such that (x,y) = a. H(y,a) contains at least onepoint x

65 Note that compactness cannotbe weakened to "closed" in 11; e.g., let

I b = 0). K = ((a,b) EE2 I a , 0, b > 1/a) and H = ((a2b)

There are two types of separationproblems implicit in the above.We may ask how strong a separation ispossible between two convex sets with given properties. Also me may assume certainconditions on one of the sets and require a certain type of separation,and ask what properties are necessary for the second set. In general, the stronger thegiven conditions on the first set, the weaker are the necessaryconditions on the second set. See Klee [3] for theorems of this type.

It should also be notedthat the separation of a flatand a convex set may be interpreted in a natural way asthe Hahn-Banach theorem.

Exercises.

if and only if 1. Show that for any convex K cEdwe have rel bd K =

K = aff K. Is the assumption that Kbe convex necessary?

2. Prove 2 and 3. mentioned in 3 be 3. If K is not convex, is itpossible that all the sets

different? Generalize, by taking repeatedclosures and relative interiors. [strictly] separated by a hyperplane if and 4. Prove that K and K may be 1 2 only if it is possible to[strictly] separate 0 and Ki - K2.

the convex sets 5. Show that 5remains valid if the assumptions on d K,KcEare weakened toeither 1 2

(i) 8(K121(2) > 0;

or (ii) K1is bounded andcl Kl n cl K2 = 0.

66 intersections of K and K with a sequence of 6. Prove 7, by cmsidering 1 2 and using 6. balls with radiitending to infinity, is a which K cEdis a closed convexset and H 7. Find examples in 0. Shaw that for everybounded supporting hyperplaneof K, but H n K = H of A wehave H n cl A0 0. set A and everysupporting hyperplane is a converse of 10. (Compare the figure.) If A 8. Prove the follawing d and if for each wE bd A there closed subset of E with int A 0 0, such that w E H,then A is exists a supportinghyperplane H of A

convex.

-1

x,y E A, z E A, v E int A, wE bd A

d subsets ofE which have nosupporting hyper- 9. Characterize those convex

planes. d set inE is the intersectionof closed 10. Prove that eachcompact convex

balls.

67 sets, then the set of all 11. If K and K are disjoint compact convex 1 2 strictly separates K and K , is an pairs (u,a) such thatH(u,a) 1 2 d-1 open subsetof S x R. d The supporting function 12. Let A denote a non-emptysubset ofE

d. h(x,A) of Ais defined for all x E E by

h(x ,A) = sup( (y,x) y E A) . I

H(A;x) = If for some non-zerox EEdwe haveh(x,A)

is a supporting hyperplaneof A in direction (z EEd 1 (z,x) = h(x,A)3

x .

Prove: and The supporting function h(x,A) is positively homogeneous

convex, that is itsatisfies

h(xx,A) = Xh (x ,A)

and h (x+y ,A) h (x ,A) + h (y ,A)

d for all X 0 and x,y E E .

(ii) If A 0 02 X 0, and x,y EEd then

h(x,y+A) = (x,y) + h(x,A)

h(x,XA) Xh(x,A)

h(x,c1 A) = h(x,A). d , then (iii) IfAland A2are non-emptyand if 002cEE

H(A1 +A2; x) = H(Al; x) +H(A2; x)

(A1 + A2) (1 H(A1 + A2 ; x) = (A1 11 (A1; x) )(1 (A2 (1 H (A2 ; ).

68 homogeneous and convex function (iv) If h(x) is any postively d closed on E such that h(0) = 0, there exists a unique non-empty

convex setK such that

h(x) = h(xX) for allx EEd.

of this section 13. Find meaningful variantsof the notions and results

relevant to spherically, orprojectively, convex sets. section if one considers con- 14. What becomes of theresults of the present

vexity in the rationald-space?

Discussion. interior" would be Hausner suggested that abetter nam for "relative and not the dimen- "absolute interior," since itdepends only on the given set Perhaps it should betreated as sion of the space inwhich it is embedded. justification for the useof the the core of the set. Apparently the only interior in word "relative" is thatthe relative interioris just the usual this space is given therelative the affine subspacedetermined by the set when

topology. induction argument usedin the Klee pointed outthat the transfinite (thus being second proof of 5could be modified to ausual induction argument countable dense subsetof more palatable toundergraduates) by considering a d d E E rather than well-orderingall the points of separation theoremsit was observed In a discussionof infinite-dimensional are not asclosely related asthey that the algebraicand topological structures problem is that hyper- are in thefinite dimensional case,and a part of the

69 The separationtheorems planes correspond tocontinuous linearfunctionals. space providedsuitable hypotheses are are still validin infinite dimensional it is not true thattwo added--indeed, they are majortools in the theory--but following example shows: Let X disjoint convex sets maybe separated, as the

all sequences x = (x,x2...) of real be the (non-complete)vector space of 1 2 Let K1 c: X be thesubset numbers, all but afinite number of which arezero. Let K2 = -X1 U(03. of all sequences whoselast non-zero numberis positive.

disjoint subsets ofX and KU K= X.Yet Then K and K are convex 1 2 1 2 eachLis topologicallydense in X. Moreover,if

x =(x12...2%20,02...) E K1

with -1 in the (n+l)th place, (i.e., xn > 0) andy =(02...0,-1,0,02..0 t > 03 c K2has x E Xi as endpoint.A similar then the halfray (x ty I example Klee gave the statement is truefor each x E K2.When presenting this and results shouldbe used to give a opinion thatinfinite-dimensional examples easily done with only alittle course broader scopewhenever this may be

extra time. amount of algebra, Killgrove and Stratopoulosobserved that a certain for any undergraduate course topology and analysis seemsto be a prerequisite discussion on the levelat which a using this material.After considerable the amount of topologyto be in- college geometry courseshould be taught and here for an ideal courseto be taught cluded, Kelly remarkedthat we are aiming The universities have animportant in- at good untversitiesby a good staff. designed for high schoolteachers. fluence on the smallcolleges and courses should keep the respectof the professiort It is importantthat geometry courses the rest of modernmathematics, since and appear as an alivesubject related to

70 schools, it will filterdown to the weaker if it is thusaccepted at the good without the feelingthat geometry is a schools in perhaps aweaker form, but curriculum. dead subject that canwell be left outof a:crowded

71 4. Convex hulls. d The spaceE is convex [andclosed], and the intersection of any family

of convex [and closed] sets is again convex[and closed]. Therefore the

follawing definitions make sense: d The convex hull conv A of a subset A of E is the intersection of

d all the convex sets inE which containA. The closed convex hull

d cl conv A of A (= E is the intersection of all theclosed convex subsets

d of E which contain A.

Clearly, ifA is bounded, so are coav A and cl conv A.

An immediate consequence of thedefinitions is

d 1. For every A (= E we have cl(conv A) = cl conv A.

Proposition 8 from the pleceding sectionimplies

2. cl conv Ais the intersection of all theclosed halfspaces

which containA.

A useful representation ofcorm Ais given by d 3. The convex hullconv A of a non-empty set A (= E is the

set of all points which may berepresented as convex combinations of points n in the form E a.x.,where of A; that is, points which can be written i=1 1 1 vn xiE Ap ai Op a= lp n = lap...

In many applications thefollowing result is very important. d 4. If Ais a compact subset of E then cony Ais closed;

in other words, for compact A wehave cl conv A = conv A.

Using the results of the precedingsection it is not hard to give a

direct proof of 4, by induction onthe dimension d. Since a much simpler

proof results fromCaratheodory's theorem, we defer the proof of 4till we

establish 5.

72 and has the basic resultsin convexity, The followingtheorem is one on fields. importantapplication in other d If A is asubset of E 2 then each 5. (Caratheodory's theorem) wherexE A, in the form x =E0. a x x E conyA is expressible i

0, E a = 1. i i=0 i xi E A, letx =EL0 aixi (with Proof. Letx E convA be gtven; combination of be a representationofxas a convex a. 0, E a =1) 1 i=0 of points ofA. smallest possiblenumber p+1 points ofA, involving the showing p d. Indeed, assuming We shall proveCaratheodory's theorem by dependent. Thus the set (x02...2x ) is affinely p d + 1 it follows that 4=0 Oixi = 0 not all equalto 02 such that there exist 01.2 0 5 i p, that generality wechoose the notation so andEP p = 0.Without loss of 0 a ai For (0 g i g p-1) for whichpi > 0. > 0 and -2. for all those i Pp Pi a an ,p-1 Li=0 yi = ai - 4=0 = 1. 0 g i g p-12 let yi = ai-r0i. Then vp pi . 0then Oi g 0 then yi 0, if Moreover, yi 0; indeed, if

a a = aixi = x 4:36yixi =ei:(1) r ,oxi yi =0,(4- 0. Thus A. Pi Pp combination of lessthanp+1 points is a representationof xas a convex of p. This completesthe proof of A2 contradicting theassumed minimality

ofCaratheodoryls theorem. Indeed, if x E ci convA, there The proof of4 is naw immediate.

thatx = lim x ByCaratheodoryts exists a sequence xE corm A such r1-41= n 0 6. X 1 for 2 wherex E A, theorem each x=Ed X x nli n 1.1=0 n2i n2i and of A guaranteesthe existence each n. The compactnessof [0,1]

and (x ) such that lim X of convergingsubsequences (X ) 10.402 mit2i nk2i

73 X(i)= 1, x(i)E A, Then obviously 0 X(i) 1 Ed and lim x =x(i). 2 i=0 Ic4c° nteL and x = E0X(i)x(i), as claimed. i= Caratheodory's', in the sensethat either is A result closelyrelated to Radon's theorem: easily derivablefrom the other, is d to find subset ofE , it is possible 6. If A is a (d+2)-pointed conv A* n convA** 0. disjoint subsets A*,A**of A such that x ) theorem is very easy. Let A = Ocol A direct proofof Radon's d 1 not affinely dependentthere existai, Since d+.2 points in d-space are d I :(1+1 Without loss of such that E a, = 0andE a.x. = O. all equal 0, i=0 i=0 so that a02..,apare positiveand generality wechoose the notation e=0ai > 0 and non-positive. Then Ogpgd.Leta= °10-12""14+1 ai The andyi = 77for p4-1 gigd+1. define $1. = -E-y. for Ogigp d+1 rewritten in theform41=0 $ixi = affine dependenceof A can be d+1 4=0 $1.= Ei=p+1 yi =1, this relation expresses Since pi 0,yi 0, and as claimedby Radon'stheorem. x ) nconv(cp+ ...x ) 0, conv(xo" P 12 d 1

Exercises, if and only f[C:Ed supports a setAc:Ed 1. Show that ahyperplane

if H supports cony A. set is compact; that the convexhull of a compact 2. Proposition 4 states is open. The convex hullof a closed show that the convexhull of an open set conv A find a closed setA 0 such that set is notnecessarily closed; whole space. is an open propersubset of the I d I x1,x2 E Al, letT(A) = T(A), 3. For Ac: E letT(A) (i(x1+x2)

74 m 1 DenoteT*(A) = U Tn(A) Shaw and for n I let T (A) =T(Tn(A)). n 1 If A = bd K although in general T*(A) cony A. thatcl T*(A) = cl conyA, d show that cl K = T(A). where K is abounded convex setinE , d 2, d E A, 0 g. X g. 13. 0(A) = (Xx+ (1-X)x I xx 4. For A c: E let 1 2 122 =8(en(A)) for n 1. Show that Define 01(A)= 0(A) and01(A) d n sets Kc: E for which cony A = U 0(A). Characterize those convex n Klee [1], SectionI. K = 0(bd K). See Bonnice - d there exists Ifx E int conyA. (= E , 5. Prove Steinitz'stheorem:

containing at most 2d points such that a subsetA*of A, general, 2dmay notbe decreased in x E int convA*. Show that thenumber in A. for which 2d points are needed and characterizethose A and x d Then x E rel int conyAif and 6. Let Ac: E be a finite set. of A, a convexcombination of allpu: only ifxis representable as

with all coefficientspositive. the sets A*and A**are unique 7. Shaw that inRadon's Theorem 6 affinely independent. Show also if and only if every d+1 points of A are A belong to the sameset if andonly if that in this casetwo points of determined by theremaining d points. they are separatedby the hyperplane if convex setsin the d-sphere, 8.What becomesof the above results rational d-space areconsidered? the projectived-space, or the the convexcombinations What are theanalogues of theabove results if (See Section 2,remark 2.) are replacedby other typesof combinations?

75 Discussion. alternate proof of4: Let Klee presentedthe following which is clearlycompact. 6:11-1-(X02". dtd)E0441X....-0,EX.,----1)

Let X denotethe compact set d+1 D x A X ... XA . d+1 times

d ao,...,ad) -0 Ei.0Xiai. ByCaratheodory's Map Xinto E .by ((X0,...2Xd), conv A, and the map isclearly ct.,...- theorem, the imageof X is exactly cony Aof the compactset X mustbe tinuous. The continuousimage

compact. proof ofCaratheodory used induction GrUnbaum remarkedthat the original needing to knowresults This had thedisadvantage of on thedimension d. while with theirsupportinghyperplanes, about the intersectionof compact sets

the presentproof is moreelementary. Caratheodory's theorem, ifp E convA and Klee pointed outthat in points, one of B c: A of at most d+1 x E A, then p E Bfor some set A x maybe thus specifiedin advance. which is x. Only one suchpoint proof gets thisstronger result. slight modificationof the given reported recentlyby Motzkin[2]. A similargeneralization was number of uses inother fields aswell Caratheodory's theorem hashad a large and the separation ofCaratheodory's theorem as convexity;for example, use proofs of someresults aboutdoubly theorems havesignificantly simplified

stochastic matrices. that if the givenset ofd+2points in Johnson andGrUnbaum mentioned specifically if thepoints are ingeneral Radon's theorem is notdegenerate, disjoint subsets way todivide them into two position, thenthere is a unique

76 whose convex hullsmeet. d of (d+1) + (k+1) points Reay mentionedthat if a set A (= E position, then it maybe divided into twodisjoint (D k d) is in general k-dimensional set.Mbre generally,if subsets whose convexhulls meet in a points, then it maybe divided into r disjoint Ahas (r-l)(c1+1) + (k+1) k-dimensional set. However, whenr > 2, subsets whose convexhulls meet in a general position isnecessaryfor the set then an independencestronger than

A. See Reay [1].

77 r"

5. Helly's Theorem.

The present section is devoted to aresult known as Helly's theorem. It

has many applications to differentfields, and it occupies a central position

among combinatorial-geometricproblems.

Let H = (K1,...,K be a finite family of convex sets in 1. (Relay) n d H, consisting of at most d+1 E , d 1. If for each subfamilyH' of

sets, there exists a point common toall members of H', then there exists

a point common toall sets ofH.

It is easy to give exampleswhich show that the convexity ofeach set

1 K. is necessary. If Kn C E is the closed ray En,a) for each integer

n, all hypotheses hold except thefiniteness ofH, yet the conclusion

fails. So H must be finite. (See 2 below.)

We shall give two proofs ofHelly's theorem; each of them exhibits

certain features which reappear inthe proofs of many combinatorial-geometric

results. Note that no generality is lost inassuming n d + 2.

The first proof is due to Helly himself.It consists of two technically

distinct steps.

(i) It is enough to prove thetheorem under the additional assumption

Indeed for each (d+l)-tuple (i i ) c: that each set is compact. 0 d

(1,...,n]letx(i0"..'id)ErldK..LetX.,for j = 1,...,n, be j=0 1 .7 i C. = conv X.. defined byX. ={x(i,.,i.. )1j E [i,...li11 and let j 0 d 0 d ' J SinceXiisafiniteset,eachC.is compact;also, X. c:K. implies

C.CK..bloreover,theintersectionofanydi-IsetsC.,...X. is non- .] J 30 Jd empty, since it contains x(j0,...,jd). But if Helly's theorem isassumed

for families of compact convex sets, wehave

78 nn K. nn C. 0 i 1.=1. i for general convexsets. Note that the hence, the validityof Helly's theorem proof. finiteness of His necessary inthis part of the of Helly's theorem forfamilies X consisting (ii) In order to prove d = 1. Indeed, it is compact convex setswe first noteits validity for

is, ford = 1, a closedsegment, possibly ilmnediatethateachK.(which1 the left endpointsof the degenerating to apoint) containsthe rightmost of theorem is false; amongthe families for members of X. Assume now that the is as small aspossible; then which it fails,choose one for which d d E for which thetheorem does not d > 1. Among all the familiesin that possible cardinality n. Then hold, select a familyH with smallest containing n-1 members, theinter- n > d + 1, and for eachsubfamily of H ,-1n Let K = H. K. 0 0; then K n Kn = 0 section of its membersis non-empty. 1=0 1 hyperplane H strictlyseparating K and and by Theorem 3.5there exists a for i = 1,...,n-1.Then 1 subset of the (d-1)-dimensional space H; each C. is a compact convex 1 a non-emptyintersection: 1/ Id has points in K as well mve0 because n1. K n = n(n K.) n.C. cj=1 j= H minimality assumption onX and the choice of as in K.Hence, by the Ci = H n(n7:1K. = H n K =0. This we have thecontradiction 0 ii

completes the firstproof of Helly'stheorem. back to Radon. We began by The second proofof Helly's theorem goes Simplifying thenotation used establishing its validityin case n = d +2.

n = d +2, in (i) above,let, for i = 1, 2,

79 jOi x. E n K.. 1 lgjgn 3 31 ,32 (= (1,...,n3 such that By Radon's Theorem4.6 there exist sets

(xi i E J23 0. conv (xi I i E J13 n conv I for each j E J2 Let xbe any point ofthis intersection;then x E K. contains contains all x. with i E J and hence K. since each such K. 1 1 Ji andJ2 we see that also their convexhull. Reversing the rolesof

0. x E K. also for all j E J1, i.e., x EnL1 Ki d all families in E consisting of Hence Helly'stheorem is valid for From a family X =(K10...21c, n g d+2sets. Now we applyinduction on n. where Ci = Ki n with n > d+2, we form afamily 0=(C1,...,Cn_11, each d+1 membersof C. is non-empty for The intersection of d+2members of X, which since it coincideswith the intersectionof some But the inductiveassumption then is non-empty bythe abave special case. and the proof iscompleted. implies 0 Ci =1=1 K it is easy to derivefrom 1 Using the standardcompactness arguments,

theorem. the following infiniteversion of Helly's d subsets ofE such that 2. If Xis any family ofcompact convex non-empty intersection,then the every d+1 or fewermembers of X have a

intersection of allmembers ofXis non-empty. Helly's theorem, withreferences An extensive list ofresults related to Danzer-GrUnbaum-Klee survey to the original papers,may be foundin the are meantonly to give listed in the bibliography.The following exercises hand at applyingeither one of the the reader theopportunity to try his in their proofs. above versions ofHelly's theorem, orthe techniques used

80 Exercises.

if the convex setsK are assumed tobe 1. Theorem 2 remains valid i (Even closed, provided somefinite subfamily has abounded intersection. omitted, but then someadditional the "finite" of thepreceding sentence may be facts--e.g., Theorem 7.3--have tobe used.) d subsets ofE such that, 2. Let Xbe a family of compact, convex or fewer membersof X have for some fixedk with1 nkNd+1, eachk

d 1 - k there a common point. Then for every flat L (=Ed of dimension L which intersectsall exists a flat L' of the same dimension,parallel to

the members ofH. d sets inE and let C (=Ed 3. Let X be a family of compact convex Xthere exists a be a compact convex set. If for every members of then there exists atranslate of translate ofC intersecting all of them, statement results also Cwhich intersects all themembers of X. A valid "containing," or by "contained if "intersecting" isreplaced throughout by

in."

A special case of thisresult was first establishedby Vincensini [1].

The following simple proofis due to Klee[2]. EachCi is Ci = (x EEd (lc C) n Ki 0). For eachKi E X, let I have a point in common. By convex, andeach d+1or fewer of theCi

Helly's theorem, there is a point x common to all Ci, so (it+ C) n Ki 0

for each K E X. inEd. Then A and 4. Let A and Bbe two finite sets of points

B may be strictlyseparated by a hyperplaneif and only if for each points, there exists a N:c: A U B such that X contains at most d + 2

hyperplane strictly separatingX n A from X n B.

81 2 inE , each 5. Let 4 be a finitefamily of parallel line segments line which intersects three of which admit a commonline transversal (i.e., a members ofZ. each of them). Then there is a linetransversal common to all

Assume with no loss ofgenerality that Z has atleast three members L E;e let CL and all segments areparallel to the Y-axis. For each segment 2 is intersected by the denote the set of all points (a,b) E E such that L such sets have a common line y = ax + b.EachCL is convex and each three

theorem there is a point (a b) E nL C The line point, so by Helly's 0'0 E,4L. y =a0 x+b isacommon transversal. 0 Using the first exercise Note that this may beeAtended in several ways. restriction of above, if all the line segmentsare closed,then the finiteness lines, we may usetransversal may be relaxed. Instead of using transversal of the given polynomials of degree n, provided we demand thateach (n+2)

segments have a commonn-polynamial as transversal. Helly's Also see Rademacherand Schoenberg [1] forother applications of

theorem. something for The general idea ofapplying Helly'stheorem is to prove objects in order to conclude small sets or fewerelements) of certain It has frequentlyhappened that it is true forthe set of all theobjects. particular problem was too that the collection ofobjects considered in a For example, if wewished small to be suitablefor an inductiveproof. 2 all elliptical domainsinE , to prove Helly'stheorem for the family of of the any attempt towork with the foci orother geometric properties

ellipse would probablyend in a disaster.

82 Discussion.

Klee mentioned that a negative form ofHelly's theorem is frequently very useful, particularlyin analysis: If Xis a family of convex sets

(with suitable restrictions) with empty intersection, then some small subfamily must already have empty intersection.

One other application of Helly's theorem isthe following (see DGK El]

d Theorem 2.5): Let X be an infinite compact set in E (a d-dimensional art gallery), and suppose that for each 6+1 points ofXthere is a point x E X from which each of these d+l points are visible (i.e., [x,uillc:X).

Then Xis starshaped (i.e., there is somex E Xfrom which each point of the art gallery is visible).

This problem leads directly to many unsolved"illumination" problems.

A famous one mentioned by Klee was: Given a triangular billard table, is there some point and some direction so that a(one-pointed) billard ball will describe a path which is dense in the triangle? Is it true that from all points there exists a direction which givessuch a path? For general triangles almost all such meaningful questions of this type areunsolved.

It was pointed out that various theorems similar to Exercise5 are known where the segments of that exercise arereplaced by various other kinds of sets. These results lead many (including Helly himself) tolook for a version of Helly's theorem that was valid in amuch more general setting.

One such generalization was given for non-convexsets and is called Helly's topological theorem.

It was noted that GrUnbaum hasgeneralized Helly's theorem in the following way (see GrUnbaum [1]): Let G be a finite family of convex sets in

d E and suppose 0 gik g J. If eachh(k) or fewer sets of have an

83 is k-dimensional, intersection which is atleast k-dimensional,theo n lc where h(0) = d + 1

h(k) = 2d + 1 - k (1 Vik d-1)

h(d) = d + 1

k = 0and Vincensini's theorem Note that this isjust Helly's theoremif

ifk = d.

84 points, faces and poonems. 6. Extreme and exposed d is an extreme point Let K be a convexsubset ofE A point x E K x = ky +(1-k)z implyx = y = z. of K provided 57,z E K, 0 < < 1, and if it does notbelong to the In other words, xis an extreme pointof K The set of all extremepoints relative interior of anysegment contained inK. x E ext Kthenx conv(Kr., (x)). of Kis denoted by ext K. Clearly, if d face of K if Let K be a convexsubset of E A set Fc:K is a exists a supportinghyperplane H of K either F = 0 orF = K, or if there the improper facesofK. The such thatF = K n H. 0 and K are called E Kis an exposed set of all facesof K is denotedby 3(K). A point x single point x, is a face point of K if theset (x), consisting of the is denoted by exp K. If K is ofK. The set of allexposed points of K F E U(K) is closed. The nota- a closed convexset, it isobvious that each will in the sequelbe used onlz forclosed tions ext K, exp K and 3(K) definition of exposedpoint involvesthe convex setsK. Note that the white the definitionof extreme "outside of K," i.e.,supporting hyperplanes, segments in K. point involves the"inside of K," i.e., the definitions: The following statementsresult at once from is a closed convexset, then 1. IfF E a(K) and ifK' c:K

F n K' E (o). thenx E ext K if and only if 2. If F E 3(K) and if x E F, ext F = F nextK. x E ext F; thus, ifF E 5(K), then 2 If K c E is the 3. For every convex K CEdwe have exp K a ext K.

convex hull of acircle and an exterior point

then the pointy of tangencyis an ex-

treme point, but not anexposed point, because

the only face of Kcontainingyis the

85 unique supporang segment [x231]. Note that z is an exposed point with a hyperplane whilex admits many suppnrtinghyperplanes. d x E K, and letBbe 4. LetKbe a closed convex set inE , let x E ext(K nB), a solid ballcentered atx. Then x E ext Kif and only if while x E exp Kif and only ifx E exp(K nB).

The next two resultsexplain the role of the extremepoints. d Then K = conv(ext K). 5. Let K be a compact convexsubset of E

Moreover, ifK = cony A then ADext K. Kc:conv(ext 102 Proof. Clearly, K:D conv(ext K). In order to establish the assertion being we use induction onthe dimension of the convexset K, assume obvious in casedim K is -12 02 or 1.Without loss of generality we

d be a line such that E= aff K. Letx E K. If x ext K, let L where obviously x E rel int(L n K). Then L n Kis a segment [y2z], point of the convex set Kthere y2z E bd K. Since through each boundary F and F of K con- passes a supportinghyperplane, there exist faces z of F and F are tainingy and z respectively. Naw, the dimensions z F = conv(ext F ) and smaller thandim K; by the inductive assumption,

(above) we havex E conv(y,z)c: F= conv(extF). Using statement 2 z z ) U conv(ext F)) c:conv(ext FU ext F) cony (FU Fz) c conv(conv(ext F z z the theorem being obvious,this conv(ext K), as claimed. The last assertion of

completes the proof of 5.

An analogous inductiveproof yields also d E , which contains no line. 6. Let K be a closed convexsubset of

Then ext K 0 0.

Regarding exposed points, wehave

d 1 H = tx E E sx2u) >ei 7. Let K c:Ed be a compact: setand let

86 be an open halfspacesuch that 11÷ nK 0 0. (where u is a unit vector)

Thenlen exp K 0 O. and denote by e the distance Proof.Let K* = n K, lety E K*, 6 =max(p(x,y-eu)J xEKnbd from y to bd11+ and by 6 the number 2 2 6 + e number satisfyingp > Letz = y pu,where p is some fixed centered at the origin,let Denoting by B the solid unit ball of 0. Then by the compactness p =infIX >0 Iz + XB:D109. Clearly, p bd(2 + pE) 0. We claim that cl K*, we have z + pB :DIC* and C = (c1 109 n existence ofapointvECnbdle,we CnbdIlt=0. Indeed, assuming the 2 2 2 2 2 20e e which implies would have 8 (p(va-eu)) = p - (0-e) 2 2 the choice of p. Therefore, C C IC* 2e g 8+ e , in contradiction to point ofz + pBand therefore but clearly eachpoint ofC is an exposed

also of cl K*and of K, as claimed. above, 8 from Section3, and 4 from Lemma 7, togetherwith 4, 5, and 6

Section 4, implyStraszewicz' theorem: d set thencl conv exp K = K. 8. If Kc:E is a compact convex K' cHK. If K' 0 K, then Indeed, letK' = el conv exp K; obviously Since the compact convexsets there exists an x EK such that x K'. there exists an openhalfspace 11+ (x) and K' may bestrictly separated, le nexp K 0 by Theorem 7,contradicting such thatle n K 0. But then

the definition ofK'.

The reader isinvited to prove set, thenexp KC extIC c cl exp K; 9. If K cEd is a closed convex [Hint: K closed, therefore, if K contains noline, thenexp K 0. d (gote that this may notbe bounded and convex inE implies K is compact.

IC = cl conv expKC cl conv cl expE:211 true in moregeneral settings.) Hence

87 ext Kc: cl exp K.] conv cl exp K = K. Thus by the second part ofTheorem 5, 3 closed, To get an example inE of a set K for whichexp K is not is a closed line segment, consider corm (C U S)where C is a circle and S

not in the planeof C, whose relativeinterior meets C.

Regarding the family 3(K) of all faces of aclosed convex set K we

have: of faces of a The intersection F = n. F. of any family [F.3 10. 1=1 1 . 1 closed convex set Kis itself a face of K. according to our definitions; Proof.IfF = 0 the assertion is true Without loss of generality, we thus we shall consideronly the case F 0. and that each Fi is a proper may assume thatthe origin 0 belongs to F, given by F = K n (x2u.) = 03 where faceaK.ThenthefaceLis 1 03. Let u.issomeunitvectorsuchthatKc(x10 1 then clearly K c:[x (xiv) 01. H = (x (x,v) = 0) where v = u ; I I 1=1 i Since OEKnH, thisimplies that H isasupportinghyperplane ofK. (c,v) = 0; Now, if x E F, then (x2ui) = 0 for all i and therefore hence xEHnK andthus Fc:H n K.On the other hand,if x E K F, 0; thus then(x0.1.)>Aforatleastorlejand0c,u.)>.3 3 xiHnK. Therefore,F=HnK and Fisaface ofK, as claimed. infinite; in this casethe face The family (F1) in Theorem 10 may be intersection of finitesubfamilies of of smallestdimension obtainable as an

(F.3equalstheintersectimaallmenabersa(F.).1 show that the followingsituation is It is easy tofind examples which F E 3(C), but F 3(K). possible: K is a compactconvex set, C E 3(K) and

Consider the example in3 above; F = [y3 and C = [x,y].

This observationleads to the followingdefinition:

88 closed convex setK provided there A set Fis called apoonem* of the for Fo = F, Fk = K, and F1..1 E 3(F1) exist sets F0,...,Fksuch that Alternatively, a set P is a poonem ofK provided i = 1,...,k.

P = K n aff P and K ',dr° is convex. of K is also a poonemof K but the By thisdefinition, each face (d-1)-dimensional poonemof converse is nottrue in general. However, each

d facets and(d-1)-poonems coincide. Kc:E is a face(facet) of K, so the set, andext F = F n ext K. Clearly, each poonemF is a closed convex and conversely,each 0-dimensional Thus each extremepoint of K is a poonem characteristics of poonemsthat makes poonem is anextreme point. One of the the poonems arerelated to them a valuabletool is justthat fact, namely, faces are related toexposed points of extreme pointsanalogously to the way be denoted byP(K). of a closed convexset K shall K. The set of all poonems from Theorem 10the analogousresult: The reader isinvited to deduce of poonems of aclosed 11Theirmersection.F=11.F.ofafamily(F.)1

convex setKis in PCK). I P(= F). 12. IfF E P(K), then P(F) = (17 EPOO PnFEP(F)and PnFE3(P). 13. IfFEa(K)and PENK), then

Exercises. exposed point. 1. A convex conehas at most one then compact convex set.Show that if dim K V. 2, 2. Let K denote a 3 such necessarily closed.Find a F:c:E ext K isclosed, butexp K is not

Hebrew word for"face." * "Poonem" Isderived from the

89 that exp K ext K cl exp K.

3. If the family 3(K) of all faces of a closed convex setK is partially ordered by inclusion, then 5(K) is a complete lattice. (For

lattice-theoretic notions see, for example, thebooks of Birkhoff orSzgsz,)

The same is true for the family P(K) of all poonems ofK. (In both cases

the greatest lower bound of a family ofelements is their intersection,)

IC, shaw 4. If Kis a closed convex set and if C is a subset of

thatC E P(K) is equivalent to each of thefollowing conditions:

(i) C is convex and for every pair x,yof points of K either the

closed segment Ex,y] is contained in C, or else the openinterval (x,y)

does not meet C.

(ii) C = K n aff C and Kr., aff C is convex.

(iii) There is a flat L for which C=KnL and Kr,,14 is convex.

(iv) There exists an x E Ksuch that C is the maximal convex subset

of K satisfying x E rel int C.

for Ogign and if FcU. F., then there exists 5.IfF.E3(K)1 0 1=1 1 CF..nesameistrueifallF.belong 11,suchthatF0 1 1 0

to P(K).

6. Let K and K be closed convex sets. Prove: 1 2

M If F.,E :a(Ki) for i = 1,2, thenF1 n F2 E 5(K1 n K2).

(LO DE F.E: P(K) for i = 1,2, thenFl n F2 E P(Kn K). 1 1 2

(iii) If F E P(K1 n K2) there exist F1 E P(K1) and 1'2 E P(K2)

such that F = Fl n F 2.

(iv) If rel int Kn rel int K0 0 and if F E 5(Kn K), there 1 2 1 2 exist F1 E 3(K1) and F2 E 3(K2) such thatF = F1 n F2.

90 (v) Find examples shawing that (iv) is nottrue if

rel int Kn rel int K= 0. 1 2

7. Let T be a non-singular projective transformation,

Ax + b Tx (c,x) + 6

+ d and letH be the open halfspace H =(x E E (c,x) + 6 > 0). Prove:

(i) If A is any subset of lethenT(conv A) = conv TA.

(ii) For every compact convex setKfor which T is permissible,

1 F E P(K)). U(TK) = (TF I F E 0(K)) andP(TIO = (TF

Discussion.

Yale pointed out that the length ofthe chain F0,...,Fk in the definition of poonem can be of length at most d+1 if Fk = Kc:Ed. On the other hand,

d [x) = F in a set in each space E there are examples of extreme points 0

K = F for which the length of each suchchain is exactly d+1. The example k 2 in 3 above shows this inE .

Klee remarked that contrary to whatmight be supposed, the relation between extreme points and exposed pointsin the boundary can be quitecomplex.

3 For example, inE there exist sets which have thefollowing properties:

1) the non-extreme boundary points aredense in the boundary,

2) the extreme but not exposed points aredense in the boundary, and

3) the exposed points are dense inthe boundary.

For a number of years it wasthought that in some reasonable sensealmost all extreme points should beexposed. In the 2-dimensional example in3 above, 2 there are only two extreme pointswhich are not exposed.And inE it is

easy to see that there canbe only countably manynon-exposed extreme points,

91 line segment containedin the since each one wouldhave to be the end of some tried to prove a higher boundary of the convexbody. Both Klee and Choquet A few years ago Corson[1] dimensional version ofthis theorem and failed. in which almost all(in a showed a 3-dimensionalexample of a convex body points fail to beexposed. sense that canbe made precise) of the extreme

For open problems inthis area, seeChoquet-Corson-Klee [1].

92 7. Unbounded convex sets.

The present section deals with someproperties of unbounded convex sets,

contains 1. A closed convex set KcEd is unbounded V and only if K a ray.

Proof.We shall consider only thenon-trivial part of the assertion.

Ed centered at Let xE K, and letS = bd Bdenote the unit sphere of 0 the origin.For each X > 0we consider theradial projection

(x0 + XS) onto xo+ S, the Px =Tr(K 11(x0 + XS)) of the compact set K (1 Since radial projection is pointxo serving as center of projection.*

XS and S, the setP is obviously a homeomorphism betweenxo + xo + X If K is unbounded, then P0 0 for every X > 0. Since K is compact. X

convex and E K, we have PCP whenever p, X. Therefore, xo X the ray n P 0.Ifyo is any porht of this intersection, X>0 X K. This completes the (Xyo +(1-X)x0 I X 03 is clearly contained in

proof of 1.

I X 03 be a ray 2. Let K cEdbe closed and convex, letL = (Xz only if emanating from the origin, andlet x,y E K. Then x+LCK if and

y + L K. be given, X 0. Proof.Letx + LCK, and let y + Xz E y + L Xz E K. Since 0 < p, < 1, consider the pointv = (1-p,)y +p,(x + For Pi y + Xz and v is eo(v ,y Xz) = (0,p,(x-y)), the distance between Pi is closed, arbitrarily small provided > 0 is sufficiently small. But K

with center of projection x * IfxoEEd, the radial projectionrr, 0) d is defined by Tr(x+x) = x + of E tx onto the unit sphere xo+S o 0 im 01

93 xandyplay equivalent and thereforev E K impliesy + Xz E K. Since 1.1 roles, the proof of 2 iscompleted. d provided -x+ C A closed convex set CcE is a cone with am x0 0

xo is pointed provided is a cone with apex 0. A cone Cwith apex The following assertions are xE ext C.Let Cbe a cone with apex 0. 0 easily verified: d C n (-C) of E . There- (i) The apices of C form a linear subspace all points of which are fore, either Cis pointed, orthere exists a line, apices ofC.

(ii) C = C + C = XCfor everyX > 0. (ii), then Conversely, if a non-emptyclosed setC CEdhas property

C is a cone with apex 0. apexxo is a cone The intersection of anyfamily of cones with common define the cone with apexxo with apex xo. Therefore it is possible to d d which spanned by a set Ac:E as theintersection of all conesinE rather important in have apexxo and contain A. Though this notion is in another construction various investigations, weshall be more interested

of cones from convex sets.

Let K be a convexset and let x E K.We define

1 x + Xy E Kfor all X 03. cc K =CY which has the origin as anapex. Lemma 2 Clearly, ccxKis a convex cone K we have ccK = cc K for all x,yE K. Thus the implies that forclosed x The convex cone cat is subscriptx is unnecessary and maybe omitted. and 2, we obtain the called the characteristic coneof K. Using Lemmas 1

following result:

94 d is a closed convex 3. If Kc: E is a closed convex set,then ccK cone; moreover, ccK 0 (0j if and only if K isunbounded.

A closed convex set Kshall be called line-freeprovided no (straight) Theorem 6.6 and the last line is contained inK. Using this terminology, part of Theorem 6.9 may beformulated as: If K is line-free, then

ext K 0 0 exp K. It is also clear that everyline-free cone is pointed.

Returning to Lemma 2, we notethat it immediately implies: If Lis a

linear subspace ofEd such thatx L C K for somex, then y+LCK decomposition theorem results: for every y E K. Therefore, the following d there exists a unique linear 4. If Kc:E is a closed convex set, is con- subspace L (=Edof maximal dimension suchthat a translate of L d subspace of E comple- tained inK. Moreover, denoting by L*any linear is a line-free set. mentary toL, we have K = L n L*), where K n L*

Some information on the structureof line-free sets is given inthe

following theorem. d line-free, closed convex set.Then 5. Let KC E be an unbounded,

bounded poonems of K. K = P ccK, where P is the union of all

Proof. We use induction onthe dimension ofK, the assertion being y E rel bd Kand obvious ifdim K = 1. Ifdim K > 1 and ifx E K, let choice is z E ccK besuch that x = y z. (Since Kis line-free, such a

possible; indeed, for anyt E ccK, t 0 0, there exists aX >0 such that y E F. If x Xt E rel bd E4) Let F be any proper faceof K such that is not bounded, the Fis bounded then F C P andx E P ca. IfF where inductive assumption anddim F < dim K imply that y au v ofF. w E ccFand v belongs to 111, the union of the bounded poonems that SinceP' c:P, cat: ccK,andccKis convex, it follows

95 x = y + z = v w + z E P' + ccK + cac:P +ccK. Since obviously

K P + ccK, this completes the proof of Theorem 5.

Since for each bounded poonem FofK, we haveext F = F n ext K and

F = conv ext F, Theorem 5 implies: d 6. Let B:c:E be a line-free, closed convex set. Then

K = ccK + conv ext K.

Exercises.

is 1. Show that Lemma 1 is valid evenwithout the assumption that K closed. d 2. If Kis anyconvex set inE , show thatx,y E rel int K implies

characteristic cone ccK cc K = cc K. Moreover,for x E rel int K the x

is closed. is a rela- 3. Shaw that the decomposition Theorem4 holds also if K

ttvely open convex set. d 4. LetB:c:E be a closed convex set; thenccKis the maximal d (with respect to inclusion) subset Tc.E with the property: For every

x E K, x T c K.

I (x,u) 0 5. Let KCEdbe a closed convex set; thenccK = EEd

(x,u) for all such uthat there exists an a with KC (z f od}.

6. Let K cEdbe a closed convex set such that 0 E rel int K.

Prove that

co 1 ccK = n (-K). n

decomposition Theorem 4, let L**denote 7. Using the notation of the

another linear subspace of E complementary toL. Shaw that L** n

96 is an affine imageof L* n K. d closed, convex set, thenthere exists a 8. If Kc: E is a line-free, dim K = 1 + dim(H n K). hyperplaneH such that H n K is compact and with apex xo, there exists 9. If K cEd is a closed pointed cone is the cone with apex a hyperplaneHsuch thatH n Kis compact and K

x spanned by H n K. o Theorems 5 and 6. 10. Prove the followingresults converse to closed convex set and if (i) If Kis an unbounded,line-free then P contains all bounded K = C + P, whereC is a cone with apex 0,

poonems of K. closed convex set and if K =C + P, where (ii) If Kis a line-free, bounded, convex set, then C is a cone with apex 0 and P 0 0 is a closed,

C = ccK and P.7.) conv(ext K). providedK = cony rel bd K. 11. A convex set Kis called reducible

Prove the followingresults: then K is the convexhull of the (i) If Kis a closed convex set

union of all irreduciblemembers of FI(K). set is either aflat or a closed (ii) Each irreducible closed convex

half-flat. d-dimensional closed convex setis homomorphic 12. Shaw that each ofEd; with one of thefollowing d+2 sets: (i) a closed halfspace 0 g k d, whereBk (ii) the product Ed-kXBkfor some k with

denotes the k-dimensional(solid) unit ball.

97 Discussion.

Since Sections 6, 7, and 8 were presented in the samelecture, many of the comments at the end of each sectionapply elsewhere as well.

There is no standard way of defining the"vertices" of an arbitrary closed convex set. If each exposed point is a ertex, then eachboundary point of the circle would be a vertex. One definition could be: a vertex of

d Kc:E is an exposed point which admits d independent supporting hyperplanes. Karlin and Shapley [1] have a discussion ofdegrees of exposure of points. A vertex to a set at a boundary pointcould be defined as a point which is the intersection of all supportinghyperplanes at the point.

Klee remarked that for a certain class ofBanach algebras with unit, the unit itself is a vertex of the unitball of the Banach algebra, and many use-

ful results can be derived from thisfact. See Bohnenblust and Karlin [1].

There are also simple examples in 2-dimensionalrational space of closed convex sets which are linearlybounded but unbounded. This type of example can give the student agood feeling for the type of complication which inthe real case arises only in the infinite dimensional spaces. This example also

shows why "compact" is needed in many of thesetheorems rather than just

"closed and bounded."In the rational plane consider all pointsbetween the

"lines" through (-1,0) and (1,0) with a slope vq. Since all lines are

98 "rational," this unbounded setis linearly bounded. In this set we can get a

is sequence ofdecreasing nested closed non-emptysets whose intersection empty.

Hausner pointed outthat each d-dimensional setmust have at least 6-1-1 exposed points. generalization of polytopes Koehler and GrUnbaum pointedout that the best quasi-polytopes, where theinter- to countably-facetedbodies was the notion of

Pbe the countable section with any polytopeis a polytope.Dmanding that restriction, since each intersection of closedhalfspaces is not a strong d is also closed convex set KC- E is such a set.A wide variety of sets closed halfspaces. obtained by allowing finiteintersections of either open or get at the geomatric con- Johnson and Prenowitzsaid it would be nice to like compactness, canbe tent of these theorems.Most topological concepts, example given above, a rayfrom expressed in geometric terms.In the rational of lack of an interior pointdoes not have to intersectthe boundary because "Any ray from an interior completeness. Perhaps a geometriccondition like Klee mentioned that point meets the set ofboundary points" shouldbe imposed. ordered field, a line-free, in any finite-dimensionalvector space over any together segmentally-closed convex set isthe convex hull of itsextreme points here appears to be "How far with its extreme rays.The more general question before you must imposefurther can you go withelementary geometrical concepts

topological properties?"

99 8. Polyhedral sets. d set provided Kis the inter- A setK C E is called a polyhedral d E . section of a finitefamily of closedhalfspaces in shared by all closed Polyhedral sets have manyproperties which are not

properties is: convex sets. One of the mostimportant of these is a face of K. 1. Each poonem of apolyhedral set K facts about polyhedralsets. Before proving Theorem1, we note a few Lete=bcEEdi(x,u.). (x.), 1gign, behalfspaces, and let

n generality we shall inthe present section K = n H..Without loss of i=1 that a maximal properface of K assume that dim K = d; we shall also say

(11+ 1 i g n) is called irredundant pro- is a facet of K. The family i I JOi Kfor eachi= 1, 2, n. videdK. .n . 1+ lgjgn j (x,ui) = aj,we have DenotingH. = bdHi=(x EEd I + 2.1f1(1-4WywhereM11n + gj g n) is irredundant,then

F. = H. fl K is a facet of K. 1 H. fl mt K. 0which, This follows at oncefrom the observationthat

irredundancy assumption. The same assump- in turn, is areformulation of the

tion also implies F. = H. n K arethe facets ofK. 3. bd K =4=1 Fi,where Fof K there exists afacet In particular,for each proper face

F. ofKsuch thatF 1 3. Then LetF. = H. n K be afacet ofK. 1 1 JOi jOi 114) = n (H. n H.). n(ngign j lgjgn

namely the intersectionof the setsH. n H., Thus F. is a polyhedral set, 1gjg.n,eachawhichiseitherlf.or ahalfspace of the(d-1) dimen-

100 is of the sional space Hi. Therefore, by Theorem 3, each facet F of F.

n H. = K n H. n H. = F. n F , f or formF = F. In rel bd (H. n = F n H. 1 j 1 1 i 1 a suitable j. Thus:

is the intersection of two 4. A facet of a facet of apolyhedral set K

facets ofK.

Now we are ready for the proofof the following result which,in view of

Theorem 6.10, clearly implies Theorem1.

5. Every poonemF of a polyhedral set Kis an intersection of facets

ofK.

Proof.We shall use induction ondim K, the assertion being obvious

By Theorem 3, there ifdim K = 1. If dim K > 1, letx E rel int F.

exists a facet F. of K such that x E Fi , i.e., Fc F.. Theorem 6.12 1 Using the inductive assumption we then implies thatF is a poonem of F .

each facet of F. since1 1 is the intersection of twofacets of K, this completes the proof ofTheorem

We mention also the followingimmediate consequence of Theorem 5.

6. If Kis a polyhedral set, thenthe family3(K) is finite.

Exercises.

1. (See Theorem 7.4 for the notation.) Show that if K is a poly-

hedral set, d ni.1n (x EE (x,u.) a. , K = 1)

then ==nbcEill(xol.)03.ni./

101 2. Show that if K is asabove, then

cc K DcEEd 0,01.) 0).

p E Ksatisfies (p,ui) = ai 3. Show that ifKis as above, and if for lgigm, and (p,ui) >. oei for m

cone K = n (x EEd i=1 of a polyhedral set is apolyhedral set, 4. Shaw that every affine map and K2are polyhedral. and thatKI + K2 is a polyhedral setprovidedK1 (permissible Find a polyhedral set Kand a projective transformationT for K) such that TKis not a polyhedral set.

102 9. Bibliographic Notes.

References for Part I.

H. F. Eohnenblustand S. Karlin of Banach algebras.Annals Geometrical propertiesof the unit sphere 1. 4

of Mathematics(2), 62(1955), 217-229.

W. Bonnice and V.Klee Mathematische Annalen,152(1963), 1. The generation of convexhulls.

1-29.

T. A. Botts Monthly, 49(1942),527-535. 1. Convex sets.American Mathematical

G. Choguet, H.Corson, and V. Klee Pacific Journal ofMathematics, 1. Exposed points of convexsets.

16(1966), 33-43.

H. H. Corson 3 whose exposed points areof the first 1. A compact convex setin E Society, 16(1965), category. Proceedings of theAmerican Mathematical

1015-1021.

L. Danzer, B.GrUnbaum, and V. Klee Proceedings of Symposiain Pure 1. Helly's theorem andits relatives.

Mathematics, Vol 7(Convexity), 101-180(1963).

B. GrUnbaum intersections of convexsets. Pacific Journal of 1. The dimension of

Mathematics, 12(1961),197-201.

P. C. Hammer Journal, 22(1955),103-106. 1. Maximal convex sets. Duke Mathematical

103 S. Karlin and L. S.Shapley Memoirs of theAmerican Mathematical 1. Geometry of moment spaces.

Society, No. 12(1953).

V. L. Klee Acta ScientiarumMathematicarum 1. The generation ofaffine hulls.

(Szeged), 24(1963), 60-81. American Journal ofMathematics, 2. The critical set of a convexbody.

75(1953), 178-188.

theorems for convex sets. (To appear.) 3. Maximal separation

T. S. Motzkin in Pure Mathematics,Vol. 7 1. Endovectors. Proceedings of Symposia

(Convexity), 361-388. Minkowski-Caratheodory theorem on convexhulls. 2. Extension of the 12(1965), (Abstract) Notices ofthe AmericanMathematical Society,

705.

H. Rademacherand I. J. Schoenberg domains andTchebycheffs' approximation 1. Helly's theorem on convex 2(1950), 245-256. problem. Canadian Journal ofMathematics

J. R. Reay (To appear.) 1. An extension ofRadon's theorem.

P. Vincensini d'un theoreme de M. J.Radon sur lesensembles de 1. Sur une extension Societe Mathematique deFrance, corps convexes. Bulletin de la

67(1939), 115-119. references of thetopics covered inPart I, For moredetailed bibliographic (in the above list)and the book Convex see theDanzer-GrUnbaum-Klee paper

104 Polytopes of GrUnbaum(listed below). various books on convexity. The material containedin Part I appears in dealing with convexityand re- A rather complete listof the available books

lated subjects is givenbelow.

Books on convexity andrelated topics

(Alexandroff), Convex polyhedra. Moscow 1950 1. A. D. Aleksandrov Berlin 1958.) (Russian). (German translation: Konvexe polyeder, of convex surfaces, 2. A. D. Aleksandrov(hlexandroff), Intrinsiczeorneta Die innere Geometrieder konvexen Moscow 1948(Russian). (German transiation:

F1Hchen, Berlin 1955.)

convexity. New York, McGraw-Hill, 3. R. V. BensonEuclidean geometry and

1966. 4 Leipzig 1916. (Reprint: New York, 4.W. Blaschke, Kreisund Kugel.

Chelsea, 1949.) Theorems and problemsof 5. V. G.Boltyanskii and I. C. Gohberg,

combinatorial geometry.Moscow 1965(Russian). konvexen KOrper. Berlin 1934. 6. T. Bonnesen and W.Fenchel, Theorie der

(Reprint: New York, Chelsea,1948.) New York, Interscience,1958. 7. H. Busemann, Convexsurfaces. London, Macmillan,1948. 8. H. S. M. Coxeter,E2221.92_221112atE.

(Second edition, New York,1963.) Application of 9. H. G. Eggleston,Problems in Euclidean space:

Convexity. New York, Pergamon,1957. Cambridge UniversityPress, 1958. 10. H. G. Eggleston,Convexity.

11. L. FejesTOth,baerun_lIler.j.inderEbeneafderKuelundimRaum,

Berlin, Springer, 1953.

105 New York, Macmillan,1964. 12. L. FejesTOth, Regular figures. Lecture notes, 13. W. Fenchel, Convex cone, setsand functions.

Princeton, 1953. New York, Wiley, 1967. 14. B. GrUnbaum, Convexpolytopes. KOrper. Basel, Birkhauser, 15. H. Hadwiger, Altesund Neues Uber konvexe

1955. Oberflgche und Iso erimetrie. 16. H. Hadwiger,Vorlesun:en Uber Inhalt

Berlin, Springer, 1957. Combinatorial geometry inthe 17. H. Hadwiger, H.Debrunner, and V. Klee

1964. plane. New York, Holt,Rinehart and Winston, Proceedings of Symposia inPure 18. V. Klee (editor)Convexity. Mathematical Society, 1963. Mathematics, Volume 7. Providence, American inequalities and related 19. H. Kuhn and A. W.Tucker (editors), Linear

systems. Princeton, 1956. Boston, Heath, 1966. 20. L. A. Lyusternik, Convexfigures and...221112pda. of Caratheodory.Memoirs 21. J. R. Reay,Generalizations of a theorem

of the AmericanMathematical Society, No.54 (1965). Cambridge, 1964. 22. C. A. Rogers, Packingand covering. New York, McGraw-Hill,1964. 23. F. A. Valentine, Convexsets. Convex figures. New York, Holt, 24. I. M. Yaglom and V.G. Boltyanskii,

Rinehart and Winston,1961. Convexity. 25. Proceedings of the1965 Copenhagen Colloquium on

1967. Kftenhavns UniversitetsMatematiske Institut,

106 Part II

The material of the second week ofthese lectures is a departure into a

more specialized areathan that of last week. The present material would not

be suitable for a year-long course inundergraduate geometry, but it would be

appropriate for a number of one-quarter courses orhonors papers at the senior

ideas level. In the second part of theselectures I will try to present more

and results, and therefore fewerproofs than last week.Much of this material

has relatively easy proofs providedthe correct sequence of provingthese

results is found. In general, since we wrkwith finite systems, less back-

ground in analysis and topology isrequired.A student can get by with only

an algebra background.

Unfortunately, the terminology inthis area has not beenstandardized.

Throughout this part of the notesthe (perhaps unbounded) intersectionof a

finite number of closed halfspaceswill be called a polyhedral set. A d-poly- d tope will always denote abounded d-dimensional polyhedral set in E .

These definitions, as wellas most of the othernotions, results and

proofs of Part II are presentedin detail in the bookGrUnbaum [1].

107 10. Number of facets ofpolytopes. for computer programshas led to As mentioned before,the timing problem d with f facets, how many the general question "Given a polytope inE five years agothe answer wasunknown vertices may ithave?" Until about ideas by Klee [23 in1962 except for certain easycases. Thanks to some new of the problem thatis the problem is nowalmost solved. The little residue still left, however, is verychallenging and strange. will use the dualprocedure, Rather than attack theproblem as stated, we polytope given itsdimension namely, find the maximumnumber of facets of a equivalent problemsbecause and the number of itsNertices.These are really Here d = dimension ofthe poly- of the existence ofduality among polytopes. (v,d) = maximum numberof facets of a tope P, v = number ofits vertices, d (analogous to many polytope inE with v vertices. To avoid trivial cases assume d 2. This keeps a problems with the emptyset) let us from now on increasing function ofv" from being result like "(v,d) is a strictly P, f(P) = the number ofk-faces of false or ill-defined.We also let k write fk. The first result is -1 k N d, and if P is understood we

immediate.

1. p(v,2) = v 3 gf0,3) = 2f0 - 4. 2. In E , f2 facts. For a proof of2, we just combinethe following

(Euler's equation) (1) fo fl f2 = 2 P is a simplicialpolytope (i.e., (2) Assume without lossof generality that

all its facets are(d-1)-simplices = triangles)

incidences of edgesand facets of simplicial3-polytopes (3) counting the

in two different ways, 2f1 = 3f2.

/d/109 From (1) and(3), f2 = 2f0 - 4 =p(f0,3). 4 we can get exampleswhich show thatthese (4) Vote that for each fo upper bounds areassumed. through these same stepsbut in general In higherdimensions we try to go always get the result. their proofs areharder, and togetherthey might not

Further they frequentlylead to unsolvedproblems. formula, the analogueof step We first turn to adiscussion of Euler's

(1) above.

. 3. Ei=-1 the formula frequentlyis Since f = f = 1 for all d-polytopes, -1 d d-1 d This formula wasknawn to &Lim' the form (-1)1 = 1 -(-1) Ei=0 published proofs of it camein the Schlafli as early as1852 but the first proofs were published,all of which early 1880's when anumber of independent

in their reasoning. (See GrUnbaum[l], were incorrect(or rather incomplete) 3 of each of theseproofs was to start with Chapter 8.) The basic idea inE edges and vertices)and note that one facet(and its associated This formula remainsvalid if a secondadjoining f - f+ f = 1holds. 0 1 2 assumed that it ispossible to always suc- facet is added. These authors such a fashion thatthe cessively add facetsofP (and their faces) in present 2-complex((d-1)-complex) in newly added facetmeets the already ((d-2)-complex). This continues, a simplyconnected polygonal linesegment d-1 i Ei=0 (-1)fi = 1) until preserving the formulafo - fl + f2 =1 (or thus giving Euler's the last facet whichmust then beadded individually, proofs is certainly notobvious, formula. The criticalassumption of these the assumption maybe established by an even for d = 3. In case d = 3

d 4 the problem is still argument usingthe Jordan curvetheorem. For

110 cast doubts onthe validity of the assump- open and infact, certain examples 3 complexes in E for which the assumptionis tion. There are simplicial correspond to a polytope.While false, though noneof these is known to 1899, the old proof Poincare seems tohave given thefirst valid proof in Most modern proofs continued to appear inbooks (e.g., Sommerville[2]). only in 1955 (Hadwiger[1]) used heavy topologicalmachinery, and it was proofs were given,and even these had and 1963 (Klee[1]) that elementary is completelyelementary and stays algebraic akrertones. The follawing proof within the frameworkof convex polygopes:

(1) Euler's equation obviously

holds for d = 1and d = 2,

thus starting theinduction

on d.

(2) It holds ifP is a

prismoid, i.e., the convex

hull of two polytopes

p and P for which 0 2' (aff P) n (aff P) n P = 0. 0 2 of a hyperplane H with To see this, let P be the intersection 1 from the pencil ofhyperplanes determined int P,where His a hyperplane f0(P2).Also if byaff Po and aff P2. Thenfo(P) = f0(P0) is either a faceof Po orP2or else 1 k g d-1, a k-face ofP corresponds to a in both P andP in which case it it has vertices 0 22 when (k-1)-face of P . Thus f fk-1(P1) 1 k(P) fk(P0) fk(P2)

1 k d-l. Summing over k we get d-1 k d1 k Ek=0 (-1) fk(P) Ek-=0(-1) (fk(P0) fk(P2))

111 By the induction applied toPi, we see that Euler'sformula holds:

d= -1 k d-1 E (-1)f(P) = 2 - (1 - (-1) ) = 1 - k0 k deduce for any d-simplex, or any (3) Euler's formula is similarly easy to d-pyramid, i.e., the convex hull of a(d-1)-polytope and a point.

(4) Let Pbe any d-polytope, and let Hbe any hyperplane which meets int Pand wtich contains exactly one vertex x of

P. LetP+=H+ nP and

P-=H- n P. If Euler's formula holds for the + d-polytopes P andP- and for the (d-l)-polytope

Po =P+n P- = H n P, then it is valid for

P =P+U P-.

The proof is analogous tothat of part (2) above when wenotice that the following formulas hold:

f(p) f0(p+) fO(P-) 2f0(P0) 1 f(p) fl(p4") f1(P-) 2f1(P0) f0(P0) + 1 and if 2 g k d-2,

fk(P) fk(Iii") fk(P-) 2fk(P0) fk-1(P0). that (5) IfP is any polytope inEd, there is a hyperplane H such

no translate of Hcontains two or more vertices ofP.

be parallel hyperplanes eachof (6) Let PcEd be given, letHi,...,Hv be the parts which meetsP in exactly one vertex,and letPl,P20...0P P are of Pinto which the H divide P. Note thatP and i 1 v-1

112 Euler's P2,...,P are prismoids. By (2) and (3) d-pyramids, while v-2 of (4), Euler'sformula holds formula holds for each P . By repeated use i v-1 forP = U P . This proves the theorem. i=1 i (2) of Theorem 2 We next turn to thereduction assumption in part

above. of facets of a d- 4. In determining themaximal possible number without loss of generalitythat polytopeP with v vertices, we may assume

P is a simplicial polytoze. establish this result isof An argument that comes tomind at once to F which is not asimplex, then F the following type: If there is a facet

of inters,::ctinghalfspaces admits a (d-2)-diagonal G. Including in the set but rotated slightly which determineP a halfspacewhich is close to F-4- For example, ifF is the top about G we will "sliceoff" a piece of P.

face of a cube, ther

becomes

number of vertices remainsthe If the amount "cutoff" is small, so that the such operations same, then thenumber of facets isincreased, and repeated

gives a simplicialpolytope and 4 is true.

However, the followingexample shows that

it is not obvious thatthis sort of thing

can be donewithout an increase in the

number of vertices.

113 approach the proofof 4 is by the useof Another (andsuccessful) way to [1]). the followingtheorem(Eggleston-Gnbaum-Klee there exists an e>0 such that if 5. For eachd-polytope P, p(P,P') < e, then fk(1") fk(P) for all k. P' is anotherd-polytope with

The Hausdorffmetric p is defined by p(P,P') = inqa > 0 P C (P'4101B) and P' c: (P4aB)3 where Bis the unit ball atthe origin. reduction to simplicialpolytopes as follows. This theoremgives us the then there are too manyvertices on onefacet. IfP is not simplicial, of the vertices awayfrom Pby an amount It is thenpossible to move one vertices remainvertices. This sufficiently small to assurethat all other the number ofvertices. The com- It pulling out" operationdoes not increase quite a lot, but we areinterested binatorial structuremight have changed

f (P') f (P) as long as and Theorem 5implies d-1 d-1 in only fd-1(1")' by less than e(P). If P' still has non- the distinguishedvertex moves "pulling out" process,at each step simplicial facets wecontinue this to by a very smallamount (with respect changing eachsuccessive polytope the number ofvertices the same,and increasing the Hausdorffmetric), keeping obtain a simplicialpolytope. (non-strictly) the numberof facets until we

This establishesTheorem 4. is small, andif Ykdenotes the It is well-knownthat if p(P,P')

then Y(P) and Y(P') are close, k-dimensional contentof the k-faces, d d that in general and Y (P') are close. Easy examples show and Y (P) d-1 d-1 3 behave similarly: in E let P the (d-2)-dimensionalcontents need not with one of itsedges "slicedoff." be a cube and P' the similar cube is a lowersemicontinuousfunction of However, it maybe proved thatYk(P)

114 and that Y(P) may be extended tothe family of all compact P for all k, k

convex setsunder preservation of this property(allowing value +00)

(Eggleston-GrUnbaum-Klee [1]). This observation leads to awealth of isoperi- For a metric type problemsconcerned with d-polytopes such asthe following. Y(P) = 1, g'ven d-polytope Pwhose k-faces have k-dimensional content k

what can be said aboutT.(P) for i k, 1 i d? In particular, no 3 which one has provedthat the convex set P in E with largest volume, for vertices, Y(P) = 1, must be a polytope and nothave an infinite number of 1 nor is themaximal volume known.

We now turn to a generalizationof the third step of theproof of

2f1 = 3f2. Theorem 2. We noted that for simplicialpolytopes in E3, For The generalization again countscertain objects in twodifferent ways.

(K) denote the number of Ogi,jgd'arldforad-polytopeK,let gi.j

incidences of an i-face ofK with a j-face. The object we count in two

different mays is

for j fixed. 2.,i=j (-1)

RLcall that when we changedthe problem from that offinding a bound on the

number of vertices to that offinding a bound on the numberof facets of a

polytope, ve mentioned that it wasthe natural duality thatmade these

problems equivalent. We again make use of thisduality. A k-face (and its In the dual faces) is itself a k-polytopewhich satisfies Euler's formula. and each setting the k-face becomes a(d-l-k)-face of the dual polytope contains the i-face of the k-face becomes a(d-l-i)-face of the dual which

0-140-face.Leth.(F) denote the number of i-facesof K incident these must satisfy with the k-face F of K. Because of this duality,

the Euler relation

115 d-1 (-1)1hi(F)=(-1)d-1

formula when k = -1, F = 0and hi(F) Note that thisis just Euler's get becomes f.(K). Using this formula we

(*) =L=j (-1)1 hi(F)

L=J. (-1)ihi(F)

d-1 =(-1) f i2 On the otherhand, count- where E indicatessummation over allj-faces F. i-faces rather thanj-faces we get ing the samething, but considering

(*) = (-1)i hi (F(i)) d-1 if L=j (-1):1)fi 2

i-faces F This last equality whereE indicatessummation over all "How many j-simplices aredetermined by comes fromanswering the question polytopes assures usthat an i-face?"Since the reductionto simplicial eachi-faceisani-simplex,itisclearthath.(F(i)) = Both of

where -1 gj g d-2. Combining these formulas arevalid for eachfixed j

them we get theequations -1 S-Cd f_10- = (**) ` " j..1.1/ `

equations in the d variables It would be nicehere if these d algebraically as afunction would allow us toobtain f fO'fl"'"fd-1 d-1 obtained under theassumption that K was a of f However they were 0' r+1 that only ---- of these equations simplicial polytope,and it turns out 2 integer in z.) We state are independent. ([z] denotes the greatest about all d-polytopesin a more this result and thecorresponding result

116 denote the vector formal form. For each d-polytopeK let f(K) f (K) = (f0 (K) ,f1 (K) , fd-l(K)) EEd.

6. The dimension of theaffine hull of the set

is d-1. (f(K) I Kis a d-polytope)

7. The dimension of theaffine hull of the set

[d/2]. (f(K)I K is a simplicial d-polytope) is dimension of the flat Note that Euler'sformula requires that the lies on the hyperplane in 6 be at most d-1, since each vector f(K) d-1 d d and a = 1 - H(u,a) c: E whereu = ) E E

Proofs of these results areomitted. are independent, Even though only abouthalf of the equations (**) It is possible tofind they will still beuseful in the followingform.

independent equations givingeach fi, i [d/2] in terms of the 4 f., i < [d/2].For example in E these equations become

f = 2f -2f 2 1 0

f = f - f 3 1 0' 5 in E the equations become

f = 4f -10f0 + 20 2 1

f = 5f -15f0 + 30 3 1

f = 2f - 6f + 12,' 4 1 0 6 while in E the equations become

f = 3f - 5f+ 5f 3 2 1 0

f = 3f -6f+ 6f 4 2 1 0

f = f - 2f+ 2f 5 2 1 0' Dehn [1] for d = 4 and 5 These equations werefirst found in 1905 by Max other results. In 1927 in a complicated way asthe by-product of some

117 d Sommerville [1] worked out similarequations in E These were ignored (even by Sommerville in his 1929book LZ]) until the1960's. In 1962 Klee [3] independently found these equationsand more besides. still While the Dehn-Sommervilleequations (**) will prove useful, they What is now needed do not allow us to find p(v,d) for large values of d. istogetsomerelationbetweenfoandf.forall 0 s i < [d/2] so that be expressed in terms of f above. As a the Dehn-Sommerville equations may o has vertices first (and apparently crude)estimate, note that each i-face least one vertex, and for two i-faces to bedifferent they must differ in at surprising sense these are so fi *(i120 ) for i sd. In a certain rather polytopes, also best possible wheni < [d/2], namely, there exist certain holds in these cases. the cyclic polytopes amongothers, for which equality equations to evaluate We will use this fact,together with the Dehn-Sommerville the cyclic polytopes following (v,d) in some cases. But first we investigate

Gale [1,2]. d by Let t EEl be a real parameterand define p(t) E E is called the moment curve p(t) =(t,t2,...,td). The set (p(t) t is realj

< t are any d+1 or more distinctvalues of t, inE . If t < t< 1 2 v d is called a cyclic then the d-polytopeC(v,d) = conv(p(t1) ""p(tv)3 c:E C(v,d) is not polytope. It may be shown thatthe combinatorial type of

dependent upon the particularchoices of the t.. number v d+1 of 8. Let C(v,d) be a cyclic po1yt6pe with any

be an vertices. For any number n gEd/2], letVn = (p(t1),...,p(tn)3 (n-1)-face arbitrary subset of n vertices of C(v,d). Then Vn determines an

0 g i <[d/2]. of C(v1d).Therefore, fi(C(v,d)) =(J.1) if that each To show this we givethe equation of ahyperplane H and note

118 half- while all othervertices p(t) lie in one open p(t. E V lies on H, ) n Consider the polynomial spacedetermined by H. 2n 2 + t 0 (t-t. = 0+ t + 2n ni=1 1) 0 1 pends only on theconstants Pi only roots of V Note thatt1,...,tnare the which determinethe points of n non-negative. Thus if the polynomial andthe polynomial is d we see that (p(t),b) -00 for all t b = E E , is t = tifor some i. ThusH(b,-00) and equality holdsif and only if camplete. the hyperplane,and the proof is polytopes aresimplicial polytopes. It is easy toshow that the cyclic hold for that theDehn-Sommerville equations This is usefulbecause it implies just the propertiesof C(v,d), it is cyclic polytopes.Furthermore, using of i-dimensionalfaces of C(v,d). possible to calculatethe number f. polytopes are formulas for the(d-1)-faces of cyclic In particular,the general if d = 2n f (C(v,d)) =v (v-11 d-1 v-n n / 2(v--1) if d = 2n 1- 1. :

equations 4 f. g(v)=( fo ) to theDehn-Sommerville InE if we apply 1 2 2

we get

f f(f-3) 2 0 0

f (f-3) 3 0 0 This gives holds for thecyclic polytopes. and Theorem 8shows that equality reasoning works problem if d = 4, and similar the completesolution to our pk(v,d) equal the maximumpossible when d = 5. To summarize: letting vertices, wehave number of k-faces in anyd-polytope with v

119 9. 113(v,4) = iv(v-3)

p2(v,4) =v(v-3)

pi(v,4) = iv(v-1)

Similarly, and fl g(f2) with 10. InE5, sincef4 = 2f1 - 6f0 + 12

v2-7v + 12. equality holding forC(f0,5), it follows that p4(v,5) = Unfortunately, Analogous equations maybe found for pk(v,5), 1 g k g 3.

We know that f2 g(fT) and this method alreadybreaks down if d = 6. for cyclic polytopes)but these fi g(f2p) are true(and that equality holds equation f = f -2f+ 2f because are not of usein the Dehn-Sommerville 5 2 1 0 For larger values of d the f. have alternating signsin this equation. get out of hand.It is possible even theDehn-Sommerville equations (**) i < [d/2], and note to solve for f ingeneralintermsofthef.for d-1 But for values of thatthecoefficientsofthef.havealternating signs. the coefficients of fi in the repre- kstrictly between [d/2] and d-1, in general that no onehas even proved f are so complicated sentation of k that they have alternatingsigns. p(v,d) = pd_1(v,d) for For this reason we nowconcentrate onfinding Dehn-Sommerville equationfor this the larger valuesof d. The appropriate separately: If d = 2n case is easierto express inthe even and odd case

n+i+1(1±1)(2n-2-0., f (-1) ill 2n-1 n n-1

and if d = 2n+1 (.1)n+14-1(2n-l-if f 2n =-1 n

fn in simplicialpolytopes with f ) if 0 i < [d/2] We also knowthat i+1 It was thislatter result that equality holding forthe cyclic polytopes.

120 the major break-throughthat lets Klee generalized in1962, thereby providing Klee's result is (see Klee[2]) us determinep(v,d) in most cases.

fs is related to f5...1 by 11. For every simplicialpolytope, Furthermore, this isindependent of the formula: (s+l)f s (f -s)fs..1. s 0 holds if and only ifthe polytope the dimension of thepolytope, and equality vertices determine ans-face). is (s+l)-neighborly (i.e., each (s+1) of counting things in two This result isproved by the above process following relation betweenfs different ways, and it maybe extended to the

and f . s-3 1--s (+1s f s( f041 )f \j/s j s-j

equality holds in theabove formulae for the for 0 g j s. In particular, (because cyclic polytopes are Ed/21- cyclic polytopeswhen0 g s < [d/2]

neighborly). result can be used toobtain an As an example, we willillustrate how this when d = 8. upper boundfor fd_l for simplicial polytopes Dehn-Sommerville equation a) f = f -3f+ 5f -5f 7 3 2 1 0 Theorem 11 for simplicialpolytopes b) 4f3 g (f0-3)f2

12 a) and b) c) f7 gf3(1-73)+ 5f1 - 5f0

(1 12 \(f0 5(f0\ 5f and assuming that d) f-31\4 I \2 I 0 o i.e., fo 15. if f 15 (1-1123) 0, o 0 equalities in e) = f(C(f 8)) if fo 15 all inequalities are 7 02 the case of cyclicpolytopes.

in d) might be negative,and It is possiblethat the term (1 - 12/(f0-3)) invalid. But this canonly happen thus make the estimatein the step to d)

121 is valid only if f ..?.-.. 15. Similar argu- if f < 15.Hence the above proof 0 0 ments work in the caseof general d provided 2 n -1 where d = 2n f 0 2 (n+1) -2 where d = 2n+1

be e2ctended to get For any particularvalue of d, these results can also

k, provided, as in the above values of f for all intermediatevalues of k d. case, that f is sufficientlylarge compared with 0 d P inE with The above argumentsshow that for all polytopes

f (P) is assumed when P is sufficiently many vertices,the maximum of d-1 when do the cyclicpolytopes a cyclicpolytope. The question now arises, polytopes? of f (for fixed f and d) over all determine the maximum value k 0 is always the case. Stated in The upper bound con ectureasserts that this the upper bound conjecture other terms, we knowthat fk(C(v,d)) g pk(v,d); lists all the cases asserts thatequality alwaysholds. The following theorem (loy an elaboration cfthe above for which this conjecturehas been established

methods, or by othermeans). pk(v,d) = fk(C(v,d)) is true at 12. The upper boundconjecture

least in thefollowing cases: v is sufficientlylarge (1) for everyk, 1 k g d-1, provided d+3 (2) for every k, lg_kgd-1, provided vg

(3) for k = d-1 provided 2 n -2 if d = 2n

(n+1)2- 3 if d = 2n + 1

(4) fork = [d/2] provided

i(n2+ 3n - 6) if d = 2n

i(n2+ 5n - 4) if d = 2n+1

122 vand k provided d 8. (5) for all (allowable) have been known since atleast the The values of pk(v,3) and pk(v,4) and 6 were establishedin 1961. first of this century. The cases for d = 5 12 were establishedin separate papersby The most importantparts of Theorem conjecture itself isdue to Klee [2] and Gale[3] in 1964. The upper bound 1957 research announcementthat it Motzkin [1] whocategorically stated in a has appeared inthe intervening was true. But since nodetailed exposition it as a conjecture. years it seemsreasonable to refel to

Discussion. of Theorem 4 isfurther com- Johnson pointed outthat the first argument simply-connected polytopes plicated by the factthat there exist non-convex He also 3 simplicial subdivisionby diagonals. inE for which there is no diagrams. (Definition: asked if Theorem 3could be provedby use of Schlegel x 0 P P is obtained bychoosing a point A Schlegel diagramof the d-polytope of P, and then projectingP sufficiently near theinterior of a facetF (d-1)-dimensional representa- radially from xonto the facet F. This gives a cellular (exceptFitself) appear in a tion of the facesofP, all of which

division of F. For example: a

2-dimensional Schlegeldiagram of 3 the cube inE is shown. (gote

that it is what yousee if you"peek

in a face,") GrUnbaum pointed out in the that thequestionable assumption

123 early proofs of Euler's formula (Theorem3) is still far from obviousin general,

even in this reduced 2-dimensional case.

GrUnbaum also presented the following related problemposed by J. Steiner more than a century ago,which shows a difficulty in the idea of"slightly moving vertices."Given a d-polytope P, can you find ad-polytope P' which

is combinatorially equivalent to P (i.e., 5(P) and5(P') are isomorphic

lattices) and which has all itsvertices on a sphere? In his 1900 book,

BrUckner stated the answer was obviously"yes" if the polytope was simplicial.

His false argument said: project P to a containing sphere,and take P' to

be the convex hull of the projections ofthe vertices. The main trouble occurs

in that you may wish that four

particular points on the sphere

determine the triangles shown

above, while in fact they

determine the triangles shown

below. This projection provides

a homeomorphism but thequestion

is whether the cells have affine

representation, and if this is made to be so, isthe resulting figure still

convex?In 1928 Steinitz [2] showed not onlythat BrUckner's proof is false,

but even proved there are lots of polytopes(simplicial ones among them) for

which there does not exist any representationwith vertices on a sphere.

Steinitz's proof is more readily explained inthe dual formulation involving

3-polytopes with an inscribed sphere. There is a wide class of 3-polytopes

which do not have any realization with an inscribed spheremeeting each face.

Start with any polytope which has more vertices thanfaces, fJr example, the

124 cube. Slice off each vertex to form v new faces. If only a little is sliced off then these new faces will be pairwisedisjoint. Assume such a polytopeP has an inscribed sphere. Each edge subtends the same angle from the two points of tangency in the adjoining faces. But since there are more "newer" facesthan "old"

the sum of the subtended angles in the new faces must be more thanthe same sum for the old feces. But this

implies that two "new" faces musthave an edge in common, acontradiction. The

dual of any such example becomes anexample where the vertices cannot be in-

scribed in a sphere.

A different approach establishesthe existence of a family ofsimple

3-polytopes which have no combinatoriallyequivalent representative inscribed 3 in a sphere. To explain this approach,consider K, a cube in E with

one vertex slicedoff. If we "peek in" the triangularface, the resulting

Schlegel diagram for this polytope isclearly seen to be

125 Also assume that Assume that this slicedcube had each vertex on asphere S. of the following special sort. the projection used toobtain the figure above was take the usual stereographic Take a point z on S above the triangle, and

projection onto the oppositesupporting plane P. This stereographic projec-

P. The quadri- tion sends circles on S (not containing z) onto circles in circle (the image of thecircle which lateral ADGF (= P is thus inscribed in a facet of P). Similarly the is the intersectionof S with the corresponding circles, as ere the pentagons quadrilaterals BDGE and CEGFare inscribed in opposite angles of anyquadrilateral ADBIJ, BECHI, AFCHJ. Using the fact that lie on the inscribed in a circle add up to 1800, it follaws that D, E, F are collinear, a sides of the triangle ABC. Thus, for example, A, D, B

contradiction to their being on a(proper) circle.This proves that no poly- sphere. tope with thecombinatorial type ofK can have its vertices on a cyclic Yale asked what themotivation was for usingthe wrd "cyclic" in

the only curve which canbe used to generate polytopes. The moment curve is not and described in terms neighborly polytopes.Another curve with this property 2n E of the form of the real parameter t is the set of all pointsin This curve was used sixty (sin t, cos t, sin 2t, cos2t,..., sin nt, cos nt). Notice that it years ago byCaratheodory in his studiesof analytic functions. 2n E Caratheodory's work is a bounded,periodic curve, i.e., it"cycles" in 1950's, when Gale[1, 2] first with cyclic polytopes wasforgotten until the complicated way, rediscovered independently describedneighborly polytopes in a simple proofs to Caratheodory's work, and then wentbeyond Caratheodory to get

126 The names are due toGale. further results oncyclic and neighborlypolytopes. community should read as Coxeter suggestedthat the mathematical

Ha b." Ha choose b"rather than over It is still an Theorem 11 of Kleeis valid for allsimplicial complexes. be extended tonon-simplicial polytopes. open problem asto whether this can

127 11. Related results and zublems.

In the last section we worked directly toward an answer to the problem

"What is In answering that question we presented many results Ild-1(v2d)?" which lead to interesting generalizations or problems on their own. Several of these are discussed in the present section.

We first consider various generalizations of Euler's formula. There are many similar equations associated with other characteristics of polytopes.

The first example we shall give is a generalization of a fact kaawn to Euclid:

2 the sum of the interior angles of an n-gon inE is (n-2)n, or (n-2)/2 full angles. Let P bead-polytope, F any k-face ofP, Ogkg d, and denote by c(F,P) the convex cone spanned by P from the vertex at the

d centroid of F. Lety(F) be the fraction of E taken up by the cone c(F,P). (Intersect c(F,P) with the unit sphere whose center is at the vertex of c(F,P) to make this defiaition precise.) In the planar case whereP is a convex polygon,

if F = P

y(F) = if F is an edge

(the interior angle at P)/2rr if F is a vertex.

For each k = 0,1,...,d, define the angle sum ak(P) by

(P) = E y (F) where the sum is taken over all k-faces F of P. k Thus the result known to Euclid could be rephrased; for all 2-polytopes P a (P) = = (f-2)/2 = a(P)- 1. This result has the following 0 0 1 1 generalization. d-1 = (-1) 3Foreverycl-polytopepPei-4(1)1..(p)i=0 al The special case d = 3was discovered by Gram in 1874. It is well-

3 known that in E 2 if all faces ofP are of the same type, then they must

128 No generalization ofthis be either triangles orquadrilaterals or pentagons. Perles and Shephard fact for higher dimensions wasknown until recently when Even if we [1] obtained results of this typeusing these angle sums ak(P).

of a particular combina- know that all facets of a4-polytope are 3-polytopes vertices of these3-polytopes torial type, there is nobound on the number of

showed the possibility ofusing in different examples. Perles and Shephard of so using certain classes of polytopes asfacets and the impossibility certain other classes. Theorem 1 is unique in It is interesting to notethat the equation of

the following sense. -1 i d-1 holds for al/ d-polytopes P, 2. Ii Edi.0(-l)pei(P) = pd(-1)

thenpo = 0/ = = pd. d-polytopes We now turn to anexample of a Euler-typeequation involving For each vertex which is a vector equationrather than a scalarequation. all hyperplanes x E P, consider the cone generatedby the outward normals to consider the content *(x) which supportP at x.As in the last example, is called the of this exterior cone. For obvious geometrical reasons 41(x) of these vertex points exterior angle of P at x. We form the vector sum

using the content of the cones asweights, and define theresulting vector

That is, S(P) to be the Steiner pointof the polytope P.

S(P) = E Ilf(x) x (summed over all 0-faces x).

The most useful propertiesof S(P) are given by dimension of the spacewhich 3. (a) S(P) depends upon neither the

(b) For all real containsP nor the locationof the origin in that space.

oe,0 and all polytopes PA,

S(uP + 00:1) = uS(P) +ps(Q)

where addition on theleft is vector addition.

129 Now each i-face0 * i d-1, is also itself a polytopeand therefore admits its own Steiner point S(F). For each i, 0 gig d-1, let us define a vector pi(P) by

pi(P) = E S(F) (summed over all i-faces Fof P).

Note that pi(P) definitely depends upon the location of theorigin and in general will not be in the polyhedronP. Yet taken together they satisfy the Euler-type vector equation

d1- d-1 4. Ei.cs(-1)1 . = (1 + (-1) )5(P).

It has recently been shownby Sallee [1] that the Steiner pointsalso have a nice valuation property, namely,when Piare polytopes andP1 U P2 is convexS(P1) + 5(P2) = S(P1 + P2) = S(P/ U P2) +S(P1 n P2).

The points S(P) have been called Steiner pointsbecause in 1843 distribution on J. Steiner proposedstudying the center of gravity of a mass

2 is proportional to a curve in E 2 where the mass at a point on the curve

the curvature at the point. The Steiner point as definedhere for a polytope

is the discrete analog of the centerof gravity of such a massdistribution.

It might be noted that if the supportfunction of a convex set is

developed in a Fourier series, then atranslation of the origin changes only

the first-order coefficients ofthe Fourier series. If the origin is trans-

lated to the Steiner point then thesefirst order coefficients will be zero.

Similarly in higher dimensions, if thesupport function is developed in

spherical harmonics of the appropriatedimension, the origin is at the

Steiner point if and only if the firstorder terms are zero. Since the

supporting functions add when convex sets areadded, it is thus reasonable

to expect thatS(P + Q) = S(P) + S(Q).

130 We next turn to a generalization of Klee's inequality (rheorem 10.11) which is valid for all simplicial complexes rather than just those simpliclal

complexes realized by simplicial polytopes. If a given simplicial complex

has f k-faces, it is natural to ask what is the best possible lower k [respectively, upper] bound on the number f of i-faces of the complex, i

when i k [respectively, i k]. This problem has been solved by 3. B.

Kruskal [1] using a rather complicated proof.We will illustrate Kruskal's

result by an example. Suppose it is knawn that in a given simplicial complex

f = 30. We write f in a certain canonical form: 4 4

£4 b In general the canonical form of f starts with (a )+() . We k k+1 choose the constants a,b,c,... as follows. Choose a as large as possible

a so that f4 -..,-i (5). Having chosen a (a = 7 in this example) choose b as

a o large as possible so that f ..( )+() Continue this process until 4 5 4

equality is reached at some stage. Thus f4 = 30 =(75)+() (131) is a

canonical representation. Then for i, the best possible bounds onfi are

( .1: ( c ) given by ...r. .. where this representation continues 1 ia+1 i/ as long as the representation of f4. (Terms like ( n.) are zero terms, a+.1

j 1.) Thus in our example,

£3 (74) (4) 51 and

f6 (77) (:) (45) (77) 1' It must be emphasized that these are the bestpossible bounds for the class of

all simplicial complexes. Even though they are bounds for the f. in terms

of f for polytopes, they are generally not the bestbounds. Nevertheless, k Kruskal's result has proved its usefulness as a tool inthe theory of polytopes

131 In by helping to establish several casesof the upper bound conjecture. in the case particular it allows us toestablish the upper bound conjecture by a rather simple d = 7, and the case d = 8 follows from the case d = 7

argument. might be extended It would be interesting toknow how Kruskal's results

when extra geometric conditions areadded. We know that we canembed 2101 topologically, or even rectilinearly,each simplicial k-complexin E k+1 E . This But suppose that asimplicial ku,complex can beembedded in obtain better bounds on adds restrictions whichconceivably can be used to

the f.. i.

Given a A problem similar to the oneconsidered in Section 10 is,

d-polytope with v vertices, hawsmall may fk be?The answer is easy

but almost nothing if d = 3, and is knawn for most casesif k = d-1, much is is knawn for intermediatevalues ofk. We first show that as

bounds on f known about the lower d-1 f d-1 as the upperbounds. For a fixed dimen-

fo > dagainst f sion d, plot d-,1

placing a dot on thegraph whenever

there is a d-polytope havingthis

number of vertices andfacets. It is . .

clear from the dualitywhich takes

f0 f into f that this graph must be 0 d-1 can be found symmetric about the diagonal.Thus the lower bounds on f wozk for from knowing the upperbounds on fd.l. This technique does not

then asking for upperbounds on fd.1 in fd..2. In the dual setting we are

132 k < d-1 terms of f the number of edges. Thus these intermediate cases for 1' do not reduce to the resultsof Section 10.

For the upper bounds onfd_l we saw that it wassufficient to look at only simplicial polytopes. It would be nice to get asimilarly restricted class of polytopes in which the lowerbounds are always assumed. Unfortunately the simplicial polytopes do nothave this property. In fact no reasonable conjecture has even been made as to aclass of d-polytopes which minimize

Nevertheless, it is still interestingto restrict our attention tothe

lower bounds on f there. A certain simplicial polytopes and ask for the k adding process which builds up simplicialpolytopes inE3 has lead to

d specific conjectures in E as to what the lawerbound for f should be for simplicial polytopes. In particular, suppose that aparticular simplicial

d minimizes f If we add polytope inE with a given number of vertices d-1' an extra vertex we mustadd at least d edges, and similarly at least (d-1)

If vk(v,d) facets. This process has lead to the lowerbound coniecture:

f for all simplicial d-polytopeswith v is the best lower bound on k vertices, then

(itd)v t, vk(v,d) for 1 k d-2 \k--1/" vd_1(v,d) = (d-1)v (d+1)(d-2).

This is easily checked in E3, but inE4 it is not at all clear thatthe

process of adding an extravertex gtves analogousresults. The lower bound

conjecture has, in fact, onlybeen proved for those caseswhere an actual Perles enumeration was made for allpolytopes with this numberof vertices. is true whenever has a few days ago reportedthat the lower bound conjecture

when vo 6., 13. The there are at most di-9 vertices, thus in particular

methods used do not seem toallow for significantimprovement.

133 rim "

One immediate application of resultsof the type of Section 10 is to the

coloring problems. The famous 4-color problem has been generalized in

different ways to higher dimensions, and also tosurfaces other than the

3 formulations of the problem do sphere inE . In the higher dimensions most

not admit upper bounds in the sensethat no finite number of colors is

sufficient to color all examples. The coloring problem generally turns out

3 to be solvable on surfaces inE other than the sphere. A minorreduction

of the 4-color problem is easily obtained,namely it is equivalent to the

coloring of the 2-faces of all simple3-polytopes (i,e., polytopes with each

vertex trivalent). 4 The duals of the cyclic polytopes in E show that there are 4-polytopes

with as many 3-faces as desired, each two having a common2-face. The Schlegel

3 diagram in E of each such 4-polytope gives an exampleof as many 3-

polytopes as desired in E3,properly meeting at their boundaries, and each

two of them mill have a 2-face in common.This was independently discovered

by Eggleston, Besicovitch, and Redo about fifteen years ago.

There is, however, at least one meaningful variantof the 4-color problem

for d-polytopes which has a non-trivial solution,namely:

5. The 2-faces of any simple d-polytopes (d 4) may be colored

using at most6d - 12 colors. (The coloring is supposed to assign different

colors to pairs of 2-faces which have a commonedge.)

An open problem in this area that appears to be verydifficult is the 3 following. It is well-knawn that maps in 2-manifoldsinE may need a large

number of colors if the genus issufficiently large. If we consider manifolds

that are not just topological, but alsolocally-affine in the sense that each

11 country" is a planar convex polygon, then there might be an upperlimit on

134 independent of the genusof the manifold.A the number of colorsnecessary, universal upper limit onthe number of very shakey guesswould be that this necessary colorsis 6.

Discussion. last two sections concerncell complexes, Since many ofthe results of the GrUnbaum is related toalgebraic topology. there was adiscussiun on how this cell is not interestedin any particular pointed out thatusually a topologist to subdivisionsand uses them as atool. decomposition, butrather quickly passes complex theorist will tend toconsider one graph or On the otherhand a graph Perhaps somethingcould be donebetween and analyze itsparticular structure. problemo are quitehard. For example, no these two positions,but many such which guaranteethat it can be reasonable conditions on a2-complex are known Each 2-complex maybe embedded embedded in a spaceof dimension lessthan 5. 4 . But it 5 manifold then it maybe embedded in E in E , and if it is also a 3 be embedded in E , would be nice toknow, for example,which 2-complexes may Note that eventhough a refine- both topologicallyand piece-wiselinearly. in geometrically as well astopologically embedded ment of a2-complex may be embedable. This is 3 that the original2-complex is so E , this does not imply where every1-complex may beembedded in contrast tothe one-dimensional case, 2 embedded geometri- 3 into E as well,then it may be inE , and if it goes

cally, that is withstraight line segments. which is the unionof 9 hexagons as As km example,consider a 2-complex topologically a torus,with the similarlylabeled shown below.This manifold is

135 that if this can vertices identified inthe obvious way. It is easy to shaw hexagons) in any be topologically and geometricallyembedded (with "flat" 3 Assuming it higher dimensional space, thenit can be embedded inE

3 is embedded in E 2 let us count the

sum of the interiorface angles of the

hexagons in two different ways. The

sum of the faceangles in each hexagon

is 417. Thus the sum of all interior

face angles is (417)9 = 3617. On the

7 other hand, 6 # of hexagons) =

3(# of vertices), by countingthe

number of edges in twodifferent ways.

Also, the sum of the interiorface

angles around a vertex is atmnst

10 Zu and equals 2ff only if the

3 hexagons at that vertexlie in a

plane. Combining these results, we

can concludethat all the hexagons

must be in oneplane, a contradiction.

136 3 12. Polytopes in E 3 Much more is known aboutpolytopes inE than about higher dimensional

polytopes, in part since it isactually possible to constructmodels of them. problem should be when there It is much easier to guesswhat the solutions of a have a simpler is an actual model tolook at. Further, 3-polytopes actually do simpler than higher dimen- structure than4-po1ytopes, just as 1-manifolds are

sional manifolds.

Part of our familaritywith 3-polytopes is due tothe possibility of

representing them in the plane. The results and methodsof graph theory may

be applied to the 1-skeleton(set of vertices and edges)of the Schlegel dia- of a 3- gram of a 3-polytope. It is clearly seenthat the Schlegel diagram

polytope is planar.The converse question was openfor a long time, namely; 2 meeting properly on theiredges which configurations in E of convex polygons

will be Schlegel diagramsof 3-poly-

topes? We want each suchconfigura-

tion to have at least fourvertices,

and each vertex must be atleast 3-

valent. Further properties are

necessary, sincethe graph in this

figure can obviously not representall

but one of the faces of a3-polytope.

of nodes (vertices) and a set In this section agraph means a finite set nodes. A graph is n-connected of edges which connectcertain pairs of these

ndisjoint paths along if for each pair x,yof vertices there are atleast disjoint if they have the edges which connectx to y. (Two such paths are Equivalently, a graph with atleast n 1 onlyxand y in common.)

137 nodes is n-connected if any n-1 or fewer nodes and their adjacent edges may be removed, and the graph remains connected. For example, the graph with solid lines is not 3-connected because the removal ofx and y and their adjacent

edges leaves u and vin separate

components° Equivalently, each path

fromu to vgoes through either

x or y, and thus there are not

three distinct such paths. However,

if the dotted line is added as an edge,

then the graph becomes 3-connected.

We say that a graph is 3-realizable \ \ if it is isomorphic to the graph of

some 3-polytope.

Another well-known property of the graph of every 3-polytope is that each such graph is 3-connected. About fifty years ago E. Steinitz gave several proofs of the converse statement unich is one of the most important and deepest results concerning 3-polytopes. Unfortunately his work was forgotten until comparatively recently.

1. (E. Steinitz [1]). Every planar, 3-connected graph is 3-realizable.

An immediate corollary is

2. Each planar 3-connected graph may be "stretched" into a shape where all its edges are straight and all bounded regions in the plane which it determines are convex polygons. (We interpret our definition of graph to exclude two distinct edges connecting a pair of vertices, and the ends of each edge must be distinct vertices.)

138 The proof of Theorem 1 is verydifficult.We will just present its main ideas and point out the difficulties.

For a graph to be 3-connected, it clearly musthave at least six edges, and it will have exactly six edges only when it is thecomplete graph on four nodes. Since we observe that this graph is 3-realizable(by the tetrahedron), we have started our induction proof onthe number of edges of the graph. Hence, assume that we are given a graph with atleast seven edges. To sketch how the proof will go, we first establish that the graph has3-valent faces or vertices.

Secondly, we will list two ways to transform a graph G into a graph G' such that ifG' is 3-realizable, then a 3-polytope P may be constructed which realizes the graph G. These transformations will be called elementary transformations. Thirdly, we will show that if a 3-connected graph Ghas a 3- valent vertex on a triangular face, then thereexists an elementary transformation ofG to a 3-connected graph G' with fewer edges than G. The induction hypothesis implies G' is 3-realizable and hence G is 3-realizable.

Fourthly, ifG does not have a 3-va1ent vertex on a triangularface, then a finite sequence of elementary transformations of G yields such a graph, in which case the above argument applies, and the proof iscomplete.

(1) Letv, e, p denote respectively the number of vertices,edges and facets of a 3-polytope P. Letvk, pk denote the number of vertices or

v = Es v and facets, respectively with exactly k incident edges. Thus k p = p , so Euler's formula may be giventhe form 3 k

4Ek3 vk 4Ew pk - 8 = 4e.

It is also clear that

Ekn kvk = 2e = Ekn kpk,

139 or

4e = Ek3 kvk kpk.

Combining these two equations we get

v3 + p3 = 8 + (k-4)(vk + pk) 8.

Thus we conclude that each 3-polytope has at leasteight 3-valent elements

(i.e., 3-valent vertices or triangular faces). Since this argument applies to

graphs on the 2-sphere as well, it also applies to planar3-connected graphs.

(2) If a 3-connected graph Ghas a triangular face, as shown on the

left below, then we define an elementarytransformation 11 on G to obtain

a graphG' = inG by removing the three edges of the triangular faceand adding

one new vertex which is then connected tothe three vertices which formad the

triangular face; if a vertex of G' is 2-valent, we replace it and the two

edges incident to it by a single edge. A typical11-transformation is shawn

belaw.

The graph GI is easily seen to be 3-connected. If GI is 3-realizable by a

3-polytope P, it is clear that the "new" vertex may be "slicedoff" by a

plane through three points corresponding tothe three old vertices, thus obtain-

ing a realization ofG. Note that if each vertex of the triangularface is at

least 4-valent, as shown above, then thenumber of edges remains the same.

However, if one or more of the vertices is3-valent, as in the example shawn

140 below, then the numberof edges may be reduced by atleast one.

ril

We also define a second typeof elementary transformation.A trivalent node and the three edges incident toit are removed from G, and the nodes con- of them are nected to it afe pairwiseconnected by "new" edges (unless some

already edges in G, in which case there is noneed for the "new" edge).

Examples of such a transformation w to get a new graph G' are shown below.

GI r,:in EacesPPP are "extended" IfP is a 3-polytopewhich realizes 123 shows a new face mightalso to get a realizationofG. As the second example obtain a need to be formed. It can be shawn thatit is always possible to

141 realization of G from any realization ofG'. In each case, an elementary transformation does not increase the number of edges.

(3) If a triangular face has at least one 3-valent vertex, then the

transformation 1reduces the number of edges. In this case the induction

completes the proof.

(4) The proof of the fact that some finite sequence of elementary trans-

formations gets us to case (3) is technical and is omitted. It uses special

properties of 4-valent, planar graphs.This completes our sketch of the proof.

Note that Theorem 1 characterizes which graphs are isomorphic tothe graphs

of 3-polytopes. It is also possible to characterize in graph theoretic terms

graphs which are isomorphic to graphs of centrally-symmetric 3-polytopes, or to

graphs of 3-polytopes with a plane of symmetry.We give one other corollary.

3. Every 3-polytope may be obtained from a tetrahedron by a finite

sequence of applications of two particular typesof operations. (Each operation

admits several different cases, as we saw abovt.,)

Tile next natural question concerns generalizations ofSteinitz's theorem

to higher dimensions. There are not even any conjectures here about what the

conditions in such a theorem should be.For example, suppose a 3-polytope is

partitioned into the union of disjoint 3-dimensional convexpolytopesyhich

meet properly along their boundaries, andwhich have appropriate connectedness

properties. It is reasonable to ask, does there exist a4-polytope for which

this is its 3-dimensional Schlegel diagram? Such configurations must have

certain obvious properties, but there exist examples whichsatisfy all such

known properties, but which are not Schlegel diagrams of any 4-polytope. The

smallest simplicial example of this consistsof just eight vertices; i.e., a

tetrahedron with only four additional interior vertices whichdefine a partition

into 19 other simplices.

142 The 4-color problem concerns the coloring of connected rlanar regions. If the graph associated with the regions is 3-connected, then Steinitz's theorem implies that it is the graph of a 3-polytope, and coloring the planar regions would be equivalent to coloring the facets of the polytope. (4 coloring assumes that regions or faces with a common edge will have different colors, and asks for the smallest number of necessary colors.) If the planar graph is not

3-connected, then easy arguments show that there are two subgraphs whose coloring would imply the coloring of the whole graph.For example, to color the graph at the left, it would suffice to color the two graphs at the right.

The corollary below therefore follows from Steinitz'stheorem, although other easier proofs exist.

4. The 4-coloring problem in the plane is equivalent to the coresponding

4-color problem for the faces of 3-polytopes.

For about 100 years, people have been trying to count the numberof

different combinatorial types of 3-polytopes withv vertices or p faces.

The problem is hard, and most early papers were attempts todraw pictures,

count them, and hope that none were missed. The problem has been significantly

advanced recently, and it seems reasonable to believe that within afew years it will be solved, at least for the simple polytopes. (Pad is simple if each

143 vertex is 3-valent.)

We now consider a simpler problem which was solved by Eberhard [1] in

1891.As above, let pk(P) be the number of k-gons in a 3-polytopeP, and

let p be the total number of facets (2-faces). We say that a sequence

of non-negative integers is 3-realizable provided there is a simple 3-polytope P such that pk = pk(P). Since for simple 3-polytopes v = v3,we have

3v = 2e = Ekz.3 kpk.

Together with Euler's formulav + p = e + 2 and

P EkW Pk we get the following theorem.

5. A necessary condition for a sequence (n 2 41) to be 3-realizable is

CIO 3p3 + 2p4 + p5 = 12 + Ek7 (k-6)pk.

It is interesting that (*) implies that there must be at least four faces with fiv e.or fewer edges, but it says nothing at all about the number of faces with six edges. Examples show that two sequences may differ only in p and yet one is 3-realizable while the other is not. This leads to the 6 following question. Given a sequence (p3,p4,p5,p7,...) which satisfies (*), does there exist a value of p which makes (p,p,p,p,p2..0 3-realizable? 6 3 4 5 6 7 Eberhard's theorem says "yes."

6. If (p304,p507,...) satisfies (*), then there is some p6 which makes the sequence 3-realizable.

The proof of Eberhard was quite hard, but by using Steinitz's theorem the proof can be simplified.Rather than prove this theorem we will prove another

144 related theorem which is easier, and yet which showsthe main idea of the proof.

If we work with 4-valent polytopes ratherthan simple polytopes we must use

4v = 2erather than3v = 2ein the above equations. Similar reasoning then yields

7. If each vertex ofP is 4-valefit, then

p3 = 8 (k-4)p . `kIt5 k

In an analogous way, this leads us to

8. (GrUnbaum) If (p32p5p6,...) satisfies (k*), then there is some p4and same 4-valent polytopeP such thatP has pk = pk (P) for all k.

The proof is accomplished by actually constructing a3-connected, planar graph, with each vertex of valence 4, and with exactly pk faces with k edges. Then Steinitz's theorem supnlies the desired 3-polytopeP. Thus we need only construct the graph. Each k-gon that we need, k 5, is formed in the following way from the inside of a square.

5-gon 6-gon 7-gon

It is clear that each such square produces onek-gon, (k-4) triangles,

and a number of quadrilaterals. Thus if we use these squares as building

blocks of our graph, we will preserve the ratiosof p3 and pk's in (**)

These building blocks are placed together asshawn on the left below. They

145 are used as the "diagonal" of a largebasic square, and corresponding edge

Basic square

points of the building blocks areconnected to the edge of the basic square as shown. We naw have a graph which is 4-valent ateach vertex except on the boundary of the basic square, and which containsexactly pk k-gonal regions.

To take care of the edge vertices, 1.eclose up the basic square with circular arcs, as shown on the rightabove. This uakes each vertex 4-valent,adds lots of quadrilaterals and exactly eighttriangular regions as shown. Thus the formula (**) is satisfied by the pk's of this graph and the proof of the theorem is complete.

A similar type of construction is applicable tothe proof of Eberhard's theorem; the details are much more invotved,mainly because of the presence of

three kinds of "small faces."No result analogous to Eberhard'stheorem is known concerning 5-valent polytopes; thee:tuation corresponding to (*) or

(**) does not have anypk missing.

146 It would be interesting to know what valuesof p can appear in 6 Eberhard's theorem, and how small the value of p6 may be. It was earlier conjectured that perhaps p6 g n where pn is the last non-zero member of the given sequence. A recent result of Barnette has disproved this conjecture.

It has not disproved a conjecture that there is a constant c such that

g c E.16 pk. Examples shaw that c must be bigger than one in this conjecture, probably the best constant is c = 3. The 1966 result of Barnette referred to is:

11. If P is a simple polytope, then

2p3 + 2p4 + 2p5 + 2p6 + p7 8 + E1,9 (k-8)pk.

In particular, this equation may be used to obtain alower bound on the size of p6. There is no similar result known for 4-valent polytopes.

Discussion.

A discussion of the proof of Steinitz's theorem brought outthe following facts:

(a) We need to apply the elementary transformation 11 to a triangular face and not a face with more sides. If Tr were the transformation shown below, for example, and ifP were a 3-polytope realizing the graphG'

147 (shown in part on the right), then there is no way to assurethat the new vertex may be"sliced off" with a plane through the four old vertices,since they may fail to be coplanar.

(b) In the fourth part,of the proof ofSteinitz's theorem, we cannot take a particular trivalent vertexand triangular face and guarantee that they can be brought together by a finite sequenceof elementary transformations.We only know that such a sequencesuffices to bring some pair of 3-valentelements together. The description of this process does, however,give an algorithm by which this can actually be accomplished in afinite number of steps; it is not

just an existence statement.

(c) Corollary 2 of Steinitz's theorem may beproved directly with less work than Steinitz's theorem, but it is not asimple fact, and false proofs have been published for it evenrecently. Direct proofs use only the graph

theoretic properties; it does not help tokeep track of which bounded regions

are convex as the inductionproceeds.Yale pointed out that this was remark-

able in itself, as it supplies thefirst known admission by GrUnbaum that con-

vexity was not a useful property.

In the discussion it was pointed outthat bounds on p in Eberhard's 6

theorem are known for certain special cases.For example, if p3 = 4, then

than or equal to 4. If p = 6, p must be zero or some even number greater 4 6 thenp6must be zero or at least 2. Some similar conditions imply thatp6

may only assumeodd values. It was also remarked that Eberhard wasblind

since his youth.

148 13. Polytopes with rationalvertices. the We now turn to a topicin integer programmingwhich extends some of following problem ideas we discussed afterSection 10. I first heard of the

does there exist a polytope P' of the from Klee. For any d-polytopeP, rational points in same combinatorialtype which has allof its vertices at

d could all the coordinatesof each vertex be E? Equivalently, we could ask, there was nothing in theliterature which integers? Until a few years ago 2 is clearly "yes" inE Is "yes" answers this. The answer to the question 2 3 dimensions 4 through 7,and is "no" but not obviously so inE , is unknown in

8 here means that the inE If we start with arational simplex (rational 3 it by the vertices have rationalcoordinates) inE and continue to transform above, elementary transformationsof Steinitz's theorem asin Corollary 12.3 each transformed polytopeis then it is no loss ofgenerality to assume that 3 using the proof of again rational. Thus the problem inE can be solved d possible to get a rationalsimplex, Steinitz's theorem. InE it is always polytope, but in general"moving or a rationalrealization of any simplicial combinatorial type of Peven if a vertex alittle" might change the

p(P,V) g e in the Hausdorff metric. example of a (convex) Perles (see GrUnbaum[1], Section 5.5) has given an This is 8-polytope with twelvevertices which has norational realization. 8 if Pc:Ed has at most the best examplepossible in E in the sense that The irrationalitiesof d+3 vertices, then it has arational realization. could realize all coordi- Perles' example are not toobad in the sense that we extension of the nates of its verticesin the field whichis the simple possible to obtainrealizations of rationals byI. More generally, it is 8 algebraic numbers. Even in E , all d-polytopes usingthe field of real

149 though, it is not known whether every polytope may be realized ubing afield which is an extension of the rationals by a finitenumber of irrationals.

Our next topic will show that this type of occurrenceis not too surprising. We will consider arrangements of lines in the projectiveplane (or arrangements of hyperplanes in projectived-space). By an arrangement we mean a set of lines in the projective plane whichdo not form a pencil at onepoint.

We are interested in questions like, what are the regionsinto which the plane

is divided, how many vertices are there, etc.We can ask when any two arrangements are combinatorially equivalent in the sensethat the equivalence pre-

serves the number of lines and verticesand their incidences. In particular we can ask if every arrangementhas an equivalent rational arrangement; i.e.,

an arrangement where thecoordinates of all vertices are rational.

In answer to the last question, the arrangementof ten lines and eleven

vertices as shown cannot have a

rational realization. It can be

shown that any arrangement combinator-

ially equivalent to this example is

already projectively equivalent to it.

Hence the cross-ratio (4,B;C,D) is

irrational in each realization. The example is actually larger than necessary.

The subarrangement of nine points and

nine lines as shown has the same

property, and is best possible in the

2 sense that any arrangement inE

with at most eight points or eight

lines has a rational realization.

150 8 Perles' example in E , referred to above, makes use of this last

d-dimen- example. His argument uses aparticular transformation which maps a sional arrangement with v verticesinto (v-d-1)-dimensional space.We "configuration," since a have used the word"arrangement" here rather than configuration is usually considered tohave a constant number of points on each line.

Discussion. 2 It was pointed out during adiscussion that the two examplesinE

which do not have rationalrealizations, can in fact berealized in the

2-dimensional vector space over thefield R.(13) (i.e., the simple extension realized if the of the rationals by1 3). On the other hand, they cannot be

field is 11(/2).

?he figure with ten lines andeleven vertices is "self-conjugate"in the

sense that anycombinatorially equivalent arrangement iseither essentially

the same or is one where thefive vertices on the points exchangeplaces with

the five vertices on the inner pentagon.

151 14. Bibliographic Notes.

References for Part II,

M. Dehn

1. Die Eulersche Formel in Zusammenhang. mit dem Inhalt in der nicht-

Euk1idischen Geometrie.Mathematische Anna1en, 61(1905), 561-586.

V. Eberhard

1. Zur Morphologie der Polyeder. Leipzig, 1891.

H. Eggleston, B. GrUnbaum, and V. Klee

1. Some semicontinuity theorems for convex polytopes and cell-complexes.

Commentarii Mathematici He1vetici, 39(1964), 165-188.

D. Gale

1. Neighboring vertices on a convex polyhedron.In Kuhn, H., and

Tucker, A. W., Linear inequalities and related systems, Princeton,

1956 ,255-263.

2. Neighborly and cyclic polytopes. Procee

Mathematics, Volume 7 (Convexity) 255-263, (1963).

3. On the number of faces of a convex polytope. Canadian Journal of

Mathematics, 16(1964), 12-17.

B. GrUnbaum

1. Convex Polytopes, New York, Wiley, 1967.

H. Hadwiger

1. Eulers Characteristik und kombinatorische Geometric. Journal fUr die

reine und an ewandte Mathematik, 194(1955), 101-110.

V. L. Klee

1. The Euler characteristic in combinatorial geometry. American Mathe-

matical Monthly, 70(1963), 119-127.

152 .1/111,,p111!11.+,,,

Canadian Journal of 2. The number of vertices of a convexpolytope.

Mathematics, 16(1964), 701-720.

Canadian Journal 3. A combinatorial analog ofPoincare's duality theorem.

of Mathematics 16(1964), 517-531.

J. B. Kruskal

1. The number of simplices in acomplex. Symposium on mathematical

optimization techniquesBerkeley, 1960, 251-278 (1963).

T. S. Motzkin Bulletin of the 1. Comonotone curves andpolyhedra (Abstract III).

American Mathematical Society,63(1957) 35.

M. Perles and G. Shepard

1. Angle inequalities .1.:or convexpolytopes. (roappear.)

T. Sallee Mt.thematika, 13(1966), 76-82. 1. A valuation property of Steinerpoints.

D. M. Y. Sommerville and volume of a polytope in 1, The relations connecting theangle-sums

space of n dimensions. aosesclinsoft (London)

Series A, 115(1927), 103-119.

2. An introduction to the geometryof ndimensions. London, 1929.

E. Steinitz

1. Polyeder and Raumeinteilungen. Encyklopádie der mathematischen

Wissenschaften, Vol. 3 (Geometrie) PartIIIAB12 (1922) 1-139.

Journal fUr 2. her isoperimetrische Probleme bei konvexenPolyedern.

reine 159(1928), 133-143.

153