Combinatorics and Convexity 1 Convex Sets in General

Combinatorics and Convexity

Gil Kalai Hebrew University Jerusalem

Connections b etween Euclidean convex geometry and combinatorics go back to Euler Cauchy

Minkowski and Steinitz The theory was advanced greatly since the s and was inuenced by the

discovery of the simplex algorithm the connections with extremal combinatorics the intro duction

of metho ds from commutative algebra and the relations with complexity theory

The rst part of this pap er deals with convexity in general and the second part deals with the

combinatorics of convex p olytop es There are many excellent surveys and collections of op en

problems I try to discuss several sp ecic topics and to zo om in on issues which I am more

familiar with

Convex sets in general

Covering Packing and Tiling

Borsuk conjectured that every b ounded set in R can b e covered by d sets of smaller

diameter Kahn and Kalai showed that Borsuks conjecture is very false in high dimensions

Here is the dispro of of Borsuks conjecture Let f d b e the smallest integer such that every

b ounded set in R can b e covered by f d sets of smaller diameter For a b ounded metric space

X let bX b e the minimum numb er of sets of smaller diameter needed to cover X Consider

d d

P the space of lines through the origin in R where the metric is given by the angle b etween

two lines The diameter of P is and the distance b etween two lines is i they are

orthogonal Let d p p a prime Frankl and Wilson see also proved that there are

d d d d

at most vectors in f g such that no two are orthogonal This yields bP

d d

since if P is covered by t sets of smaller diameter each such set contains at most of the

d d d

lines spanned by the vectors in f g But there are such lines and therefore t

d d d

Now emb ed P into R by the map x x x where x is a vector of norm in R Note

that x x y y x y Therefore the order relation b etween distances is preserved and

d d

the image of P is the required counterexample This example gives f d for suciently

large d

Betke Henk and Wills proved for suciently high dimensions Fejes Toths sausage conjecture

They showed that the minimum volume of the convex hull of n nonoverlapping congruent balls in

R is attained when the centers are on a line

Keller conjectured that in every tiling of R by cub es there are two cub es which share a

complete facet Lagarias and Shor showed this to b e false for d They used a reduction

to a purely combinatorial problem which was found by Corradi and Szabo

If x x x x and y y y y you can regard x y as the d k matrix whose i j entry is x y

1 2 d 1 2 k i j

Some problems

There are many problems on packing covering and tiling and the most famous are p erhaps the

sphere packing problem in R and the asymptotic sphere packing problem in R There are several

op en problems around Borsuks problem What is the asymptotic b ehavior of f d What is the

situation in low dimensions What is the b ehavior of bP Witsenhausen conjectured see

that if A is a subset of the unit sphere without two orthogonal vectors then v ol A v where

n n

v is the volume of a spherical cap of radius This would imply that bP o

Perhaps the algebraic metho ds used for the FranklWilson theorem can b e of help

Schramm proved an upp er b ound f d sd o He showed that every

set of constant width can b e covered by sd smaller homothets Bourgain and Lindenstrauss

proved the same b ound by covering every b oundede set by sd balls of the same diameter Danzer

already showed that exp onential numb er of balls is sometimes necessary In his pro of Schramm

related the value of f d with another classical problem in convexity that of nding or estimating

the minimal volume of Euclidean and more generally spherical sets of constant width

It is not known if there are sets of constant width in R whose volume is exponential ly smaller

than the volume of a ball of radius Perhaps the following series of examples suggested by

Schramm K R will do but we do not know to compute or estimate their volumes K

and K is obtained as follows Consider K as sitting in the hyp erplane given by x in

d d d

R Now take K A B where A is the set of all p oints z with x such that

d d d d d

the ball of radius around z contains K and B is the set of all p oints z with x which

d d d

b elong to every ball of radius which contains K Schramm also conjectured that the minimal

volume of a spherical set of constant width is obtained for an orthant

Finally what is the minimal diameter d such that the unit nball can b e covered by n sets

of diameter d It is known that O log nn d O n see Hadwiger conjectured

n n

that the upp er b ound which corresp onds to the standard symmetric decomp osition of the ball to

n regions is the truth Perhaps also here the natural conjecture is false

Hellytyp e theorems

Tverb ergs theorem

Sarkaria found a striking simple pro of of the following theorem of Tverb erg

Every d r points in R can be partitioned into r parts such that the convex hul ls

of these parts have nonempty intersection

He used the following result of Barany Let A A A be sets in R such that x

conv A for every i Then it is possible to choose a A such that x conv a a a

i i i d

To prove this consider the minimal distance t b etween x and such conv a a a and show

that if t one of the a s can b e replaced to decrease t

Now consider m d r p oints a a a in R and regard them as p oints in

V R whose sum of co ordinates is Sarkarias idea was to consider the tensor pro duct V W

where W is a r dimensional space spanned by r vectors w w w whose sum is zero

Next dene m dimV U sets in V U as follows

A fa w a w a w g

i i i i r

Note that is in the convex hull of each A and by Baranys theorem conv fa w a

i i

w a w g for some choices of i i i The required partition of the p oints is given

i m i m

by fa i j g j r To see this write a w where the co ecients

j k k k k i k

are nonnegative and sum to Deduce that the vectors v a j r are all equal

j k k

and so are the scalars

k j

There are many b eautiful problems and results concerning Tverb ergs theorem see Top o

logical versions were found for the case where r is a prime and were extended to derive colored

versions of Tverb ergs theorem Sierksma conjectured see that the numb er of Tverb erg

d d r

partitions is at least r For a nite set A in R let f A r maxfdim conv g where

the maximum is taken over all partitions of A

jAj

Conjecture f A r Note dim

This extension of Tverb ergs theorem was proved by Kadari for planar sets

The HadwigerDebrunner Piercing Conjecture

Alon and Kleitman proved the HadwigerDebrunner Piercing Conjecture

For every d and every p d there is a c cp d such that the fol lowing holds For

every family H of compact convex sets in R in which any set of p members of the family contains

a subset of cardinality d with a nonempty intersection there is a set of c points in R that

intersects each member of H

Hellys theorem asserts that cd d and it is not dicult to see that cp p

We describ e the pro of for the rst typical case d p We are given a family of n planar

convex sets and out of every four sets in the family we can nail three with a p oint We want

to nail the entire family with a xed numb er of p oints The rst step is to show that there is

a way to nail a constant fraction indep endent from n of the sets with one p oint This follows

from a fractional Helly theorem of Katcalski and Liu A more sophisticated use of the Katcalski

Liu theorem shows that for every assignment of nonnegative weights to the sets in the family we

can nail with one p oint sets representing a constant prop ortion of the entire weight Using linear

programming duality Alon and Kleitman pro ceeded to show that there is a collection Y of p oints

their numb er may dep end on n such that every set in the family is nailed by a constant fraction

of the p oints in Y The nal step replacing Y with a set of b ounded cardinality which meets all

the sets in the family is done using the theorems of Barany and Tverb erg mentioned ab ove

Convex p olytop es

Polytop es spheres and Steinitz theorem

Convex p olytop es are among the most ancient mathematical ob jects of study The combinatorial

theory of p olytop es is the study of their facestructure and in particular their face numb ers There

is also a develop ed metric theory of p olytop es problems concerning volume width sections pro

jections etc and arithmetic theory lattice p oints in p olytop es These three asp ects of convex

p olytop es are related and some of the algebraic to ols mentioned b elow are relevant to all of them

A convex ddimensional p olytop e briey a dp olytop e is the convex hull of a nite set of

p oints which anely span R A nontrivial face of a dp olytop e P is the intersection of P with a

supp orting hyp erplane The empty set and P itself are regarded as trivial faces faces are called

vertices faces are called edges and d faces are called facets The set of faces of a p olytop e

is a graded lattice Two p olytop es P and Q are combinatorial ly isomorphic if there is an order

preserving bijection b etween their face lattices P and Q are dual if there is an order reversing

bijection b etween their face lattices

Simplicial p olytop es are p olytop es all whose prop er faces are simplices Duals of simplicial

p olytop es are called simple p olytop es A dp olytop e P is simple i every vertex of P b elongs to d

edges Denote by f P the numb er of ifaces of P The vector f P f P f P is called

i d

the f vector of P Eulers famous formula V E F is the b eginning of a rich theory on

facenumb ers of convex p olytop es and related combinatorial structures

The wild b ehavior for d

The b oundary of every simplicial dp olytop e is a triangulation of a d sphere but there are

triangulations of d spheres which cannot b e realized as b oundary complexes of simplicial

p olytop es for d Go o dman and Pollack proved that the numb er of combinatorial typ es

of p olytop es is surprisingly small The numb er of dp olytop es with vertices in any

dimension is b ounded ab ove by while the numb er of triangulations of spheres with

69222525

This is achieved for d There are combinatorial typ es vertices is b etween

of convex p olytop es that cannot b e realized by p oints with rational co ordinates and there

are p olytop es which have a combinatorial automorphism which cannot b e realized geometrically

and whose realization space is not connected Mnev showed that for every simplicial complex

C there is a p olytop e whose realization space is homotopy equivalent to C Recently Richter

announced that all these phenomena o ccur already in dimension that all algebraic numb ers are

needed to co ordinatize all p olytop es and that there is a nonrational p olytop e with vertices

The tame b ehavior for d

All these pathologies do not o ccur for p olytop es by a deep theorem of Steinitz asserting that

every connected planar graph is the graph of a p olytop e and related theorems Relatives of

Ko eb eAndreevThurston circle packing theorem provide new approach to Steinitz theorem see

Andreev and Thurston proved that there is a realization of every p olytop e P such that all

its edges are tangent to the unit ball and this realization is unique up to pro jective transforma

tions preserving the unit sphere Schramm observed that by cho osing the realization so that the

hyp erb olic center of the tangency p oints of edges with the unit sphere is at the origin you get the

following result Which answers a question of Grun baum and extends a result of Mani

Let P be a polytope and let be the group of combinatorial isomorphisms of the pair P P

where P is the dual of P In other words each element of is either a combinatorial automor

phism of P or an isomorphism from P to P Then there is a realization of the polyhedron so that

every element of is induced by a congruence

An op en problem of Perles is whether every combinatorial automorphism of a centrally sym

metric dp olytop e P is centrally symmetric if x P implies x P satises v v

Face numb ers and hnumb ers of simplicial p olytop es

The upp er b ound theorem and the lower b ound theorem

Motzkin conjectured in and McMullen proved in the upper bound theorem Among

all dp olytop es with n vertices the cyclic p olytop e has the maximal numb er of k faces for every

k The cyclic dp olytop e with n vertices is the convex hull of n p oints on the moment curve

xt t t t Cyclic dp olytop es have the remarkable prop erty that every set of k vertices

determines a k face for k d

Klee proved in the assertion of the upp er b ound theorem when n is large wrt d for

arbitrary Eulerian complexes namely d dimensional simplicial complexes such that the link of

every r face has the same Euler characteristics as a d r sphere The assertion of the upp er

b ound theorem for arbitrary Eulerian complexes even manifolds is still op en

Bruc kner conjectured in and Barnette proved in the lower bound theorem The

minimal numb er of k faces for simplicial dp olytop es with n vertices is attained for stacked p oly

top es Stacked p olytop es are those p olytop es built by gluing simplices along facets

The g conjecture

Let d b e a xed integer Given a sequence f f f f of nonnegative integers put

f and dene hf h h h by the relation

P P

d d

dk dk

h x f x

k k

If f f K is the f vector of a d dimensional simplicial complex K then hf hK

is called the hvector of K The hvectors are of great imp ortance in the combinatorial theory of

simplicial p olytop es The upp er b ound theorem and the lower b ound theorem have simple forms

ndk

in terms of the hnumb ers The upp er b ound theorem follows from the inequality h

The lower b ound theorem amounts to the relation h h The DehnSommerville relations for

the face numb ers of simplicial p olytop e assert that h h

k dk

In McMullen prop osed a complete characterization of f vectors of b oundary complexes of

simplicial dp olytop es McMullens conjecture was settled in Billera and Lee proved the

suciency part of the conjecture and Stanley proved the necessity part Recently McMullen

found an elementary pro of of the necessity part of the g theorem

McMullen conjecture now called the g theorem asserts that h h h is the hvector of

a simplicial dpolytope if and only if the fol lowing conditions hold a h h b there is a

i di

graded standard algebra M M such that dimM h h for i d A graded

i i i i

algebra is standard if it is generated as an algebra by elements of degree

The second condition was originally given in purely combinatorial terms which is equivalent

to the formulation given here by an old theorem of Macaulay In the rest of this section we will

describ e metho ds used to attack the upp er and lower b ound theorems and the g conjecture

It is conjectured that the assertion of the g theorem applies to arbitrary simplicial spheres

Shellabili ty and the hvector

A shel ling of a simplicial sphere is a way to intro duce the facets maximal faces one by one so

that at each stage you have a top ological ball until the last facet is intro duced and you get the

entire sphere Let P b e a simplicial p olytop e and let P b e its p olar which is a simple p olytop e

A shelling order for the facets of P is obtained simply by ordering the vertices of P according to

some linear ob jective function The numb er h has a simple interpretation as the numb er of

vertices v of P of degree k where the degree of a vertex is the numb er of its neighb oring vertices

with lower value of the ob jective function Switching from to we get the DehnSommerville

relations h h Including the Euler relation for k

k dk

We are ready to describ e McMullens pro of of the upp er b ound theorem in a dual form

Consider a linear ob jective function which gives higher values to vertices in a facet F than to all

other vertices to obtain that h F h P Next

k k

h F k h P d k h P

k k k

where the sum is over all facets F of P To see this note that every vertex of degree k in P has

degree k in k facets containing v and degree k in the remaining d k facets and gives

ndk

by induction on k the upp er b ound relations h

CohenMacaulay rings

Stanley see proved the upp er b ound theorem for arbitrary simplicial spheres using the theory

of CohenMacaulay rings Let K b e a d dimensional simplicial complex on n vertices x x

The facering RK of K is the quotient Rx x x I were I is the ideal generated by nonfaces

x x of K Namely I is generated by monomials of the form x x x where x is

i i i i i i

m m

2 1 2 1

not a face of K RK is a CohenMacaulay ring if it decomp oses into direct sum of translation

of p olynomial rings as follows there are elements of RK and such

d t

that

RK R

i d

It turns out that the s can b e chosen as linear combinations of the variables and when this is

the case the numb er of s of degree i is precisely h Reisner found top ological conditions for the

CohenMacaulayness of RK which imply that RK is CohenMacaulay when K is a simplicial

sphere All this implies the upp er b ound inequalities for the h numb ers since roughly after mo ding

out by d linear forms the dimension of the space of homogeneous p olynomials of degree k from

ndk

which the s are taken is

Toric varieties

For every rational dp olytop e P one asso ciates an algebraic variety T P of dimension d If P has

n vertices v v v then consider n complex variables z z and replace each ane relation

n n

P P Q

with integer co ecients n v where n by the p olynomial relation z When

i i i

P is simplicial Danilov proved that the ith Betti numb er of T P is h This enabled Stanley

to prove the necessity part of the g conjecture via the HardLefschetz theorem for T P

Rigidity and stresses

Let P b e a simplicial dp olytop e d then P is rigid Namely every small p erturbation of the

vertices of P which do es not change the length of the edges of P is induced by an ane rigid motion

of R The rigidity of simplicial p olytop es follows from Cauchys rigidity theorem which asserts

that if two combinatorially isomorphic convex p olytop es have pairwise congruent faces then they

are congruent It follows also from Dehns innitesimal rigidity theorem for simplicial p olytop e

There is a simple inductive argument on the dimension to prove rigidity of simplicial dp olytop es

starting with the case d If P is a simplicial dp olytop e with n vertices there are dn degrees

of freedom to move the vertices and the dimension of the group of rigid motions of R is

Therefore the rigidity of P implies the lower b ound inequality f P dn This observation

gives also various extensions of the lower b ound theorem see Kalai Lee extended this

idea to higher hnumb ers and found relations to the facering

The algebra of weights

A remarkable recent development is McMullens elementary pro of of the necessity part of the g

conjecture McMullen proved in fact the assertion of the the HardLefschetz theorem and

his pro of applies to nonrational simplicial p olytop es There the toric varieties do not exist but

the assertion of the HardLefschetz theorem in terms of the facering still makes sense McMullen

denes r weights of simple dp olytop es to b e an assignment of weights w F to each r face F such

that in each r face G w F u where the sum is taken over all r faces F of G and

F G

u is the outer normal of F in G Let P denote the space of r weights of the p olytop e P

F G r

A well known theorem of Minkowski asserts that assigning to an r face its r dimensional volume is

an r weight These sp ecial weights have a central role in the pro of

McMullens pro of pro ceeds in the following steps He denes an algebra structure on weights

and show that this algebra is generated by weights He proves that dim P h P

r r

He considers the sp ecial weight which assigns to each edge its length and proves that

is an isomorphism To show this McMullen computes the signature of the

r dr

quadratic form x on P This is achieved via new geometric inequalities of Brunn

Minkowski typ e

Algebraic shifting

Algebraic shifting intro duced by Kalai in is a way to assign to every simplicial complex K an

auxiliary simplicial complex K of a sp ecial typ e The vertices of K are v v v and the

v v r faces of K resp ect a certain partial order Namely if S v form an r face of

i i i

1 0

K then if one of the vertices v of S is replaced with a vertex v with i j this results also with

j i

a face of K For example if v v is a face of K then so is v v The denition of

K is given by a certain generic change of basis for the co chain groups of K see

Various combinatorial and top ological prop erties of simplicial complexes are preserved by the

op eration K K K has the same f vector as K K also have the same Betti numb ers

as K but other homotopical information is eliminated as K has the homotopy typ e of a wedge

of spheres K has the CohenMacaulay prop erty its facering is CohenMacaulay i K has

What is still missing is the relation of algebraic shifting with emb eddability in R It is a well

known fact that K the complete graph with ve vertices cannot b e emb edded in the plane More

generally vanKamp en and Flores proved that the r skeleton of the r simplex cannot

b e emb edded in R Kalai and Sarkaria prop ose

r r

Conjecture is not contained in K whenever K is embeddable in R

This conjecture would imply the assertion of the g theorem for arbitrary simplicial spheres