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Home , Convex geometry

Convex Introduction

Martin Henk TU Berlin Winter semester 2014/15 webpage CONTENTS i

Contents

Preface ii WS 2014/15

0 Some basic and convex facts1

1 Support and separate5

2 Radon, Helly, Caratheodory and (a few) relatives9

3 The multifaceted world of polytopes 11

4 The space of compact sets / convex bodies 17

5 A glimpse on mixed volumes 19

Index 25 ii CONTENTS

Preface

The material presented here is stolen from different excellent sources:

• First of all: A manuscript of Ulrich Betke on convexity which is partially based on lecture notes given by Peter McMullen.

• The inspiring books by

– Alexander Barvinok, ”A course in Convexity” – G¨unter Ewald, ”Combinatorial Convexity and ” – Peter M. Gruber, ”Convex and ” – Peter M. Gruber and Cerrit G. Lekkerkerker, ”Geometry of Num- bers” – Jiri Matousek, ”Discrete Geometry” – Rolf Schneider, ”Convex Geometry: The Brunn-Minkowski Theory” – G¨unter M. Ziegler, ”Lectures on polytopes”

• and some original papers

!! and they are part of lecture notes on ”Discrete and Convex Geometry” jointly written with Maria Hernandez Cifre but not finished yet. Some basic and convex facts 1

0 Some basic and convex facts

n  0.1 Notation. R = x = (x1, . . . , xn)| : xi ∈ R denotes the n-dimensional Pn Euclidean space equipped with the Euclidean inner product hx, yi = i=1 xi yi, n p x, y ∈ R , and the Euclidean norm |x| = hx, xi.

0.2 Definition [Linear, affine, positive and convex combination]. Let m ∈ n N and let xi ∈ R , λi ∈ R, 1 ≤ i ≤ m. Pm i) i=1 λi xi is called a linear combination of x1,..., xm. Pm Pm ii) If i=1 λi = 1 then i=1 λi xi is called an affine combination of x1, ..., xm. Pm iii) If λi ≥ 0 then i=1 λi xi is called a positive combination of x1,..., xm. Pm Pm iv) If λi ≥ 0 and i=1 λi = 1 then i=1 λi xi is called a convex combination of x1,..., xm.

n n v) Let X ⊆ R . x ∈ R is called linearly (affinely, positively, convexly) dependent of X, if x is a linear (affine, positive, convex) combination of finitely many points of X, i.e., there exist x1,..., xm ∈ X, m ∈ N, such that x is a linear (affine, positive, convex) combination of the points x1,..., xm.

n 0.3 Definition [Linearly and affinely independent points]. x1,..., xm ∈ R are called linearly (affinely) dependent, if one of the xi is linearly (affinely) de- pendent of {x1,..., xm}\{xi}. Otherwise x1,..., xm are called linearly (affinely) independent.

n 0.4 Proposition. Let x1,..., xm ∈ R .

x1 xm n+1 i) x1,..., xm are affinely dependent if and only if 1 ,..., 1 ∈ R are linearly dependent.

ii) x1,..., xm are affinely dependent if and only if there exist µi ∈ R, 1 ≤ Pm Pm i ≤ m, with (µ1, . . . , µm) 6= (0,..., 0), i=1 µi = 0 and i=1 µi xi = 0.

iii) If m ≥ n + 1 then x1,..., xm are linearly dependent.

iv) If m ≥ n + 2 then x1,..., xm are affinely dependent.

0.5 Definition [Linear subspace, affine subspace, cone and ]. n X ⊆ R is called

n i) linear subspace (set) if it contains all x ∈ R which are linearly dependent of X,

n ii) affine subspace (set) if it contains all x ∈ R which are affinely dependent of X, 2 Some basic and convex facts

n iii) (convex) cone if it contains all x ∈ R which are positively dependent of X,

n iv) convex set if it contains all x ∈ R which are convexly dependent of X.

n n 0.6 Notation. C = {K ⊆ R : K convex} denotes the set of all convex sets n in R . The empty set ∅ is regarded as a convex, linear and affine set.

n 0.7 Theorem. K ⊆ R is convex if and only if

λx + (1 − λ) y ∈ K, for all x, y ∈ K and 0 ≤ λ ≤ 1.

 n 0.8 Example. The closed n-dimensional ball Bn(a, ρ) = x ∈ R : |x − a| ≤

ρ with centre a and radius ρ > 0 is convex. The boundary of Bn(a, ρ), i.e.,  n x ∈ R : |x − a| = ρ is non-convex. In the case a = 0 and ρ = 1 the ball Bn(0, 1) is abbreviated by Bn and is called n-dimensional unit ball. Its boundary is denoted by Sn−1.

n T n 0.9 Corollary. Let Ki ∈ C , i ∈ I. Then i∈I Ki ∈ C .

0.10 Definition [Linear, affine, positive and , ]. n Let X ⊆ R . i) The linear hull lin X of X is defined by \ lin X = L. n L⊆R ,L linear, X⊆L

ii) The affine hull aff X of X is defined by \ aff X = A. n A⊆R ,A affine, X⊆A

iii) The positive (conic) hull pos X of X is defined by \ pos X = C. n C⊆R ,C convex cone, X⊆C

iv) The convex hull conv X of X is defined by \ conv X = K. n K⊆R ,K convex, X⊆K

v) The dimension dim X of X is the dimension of its affine hull, i.e., dim aff X. Some basic and convex facts 3

n 0.11 Theorem. Let X ⊆ R . Then ( m m ) X X conv X = λi xi : m ∈ N, xi ∈ X, λi ≥ 0, λi = 1 . i=1 i=1 0.12 Remark. i) conv {x, y} = λ x + (1 − λ) y : λ ∈ [0, 1] . Pm ii) lin X = i=1 λixi : λi ∈ R, xi ∈ X, m ∈ N . Pm Pm iii) aff X = i=1 λixi : λi ∈ R, i=1 λi = 1, xi ∈ X, m ∈ N . Pm iv) pos X = i=1 λixi : λi ∈ R, λi ≥ 0, xi ∈ X, m ∈ N .

0.13 Definition [(Relative) interior and (relative) boundary point]. n Let X ⊆ R . i) x ∈ X is called an interior point of X if there exists a ρ > 0 such that Bn(x, ρ) ⊆ X. The set of all interior points of X is called the interior of X and is denoted by int X.

n ii) x ∈ R is called boundary point of X if for all ρ > 0, Bn(x, ρ) ∩ X 6= ∅ n and Bn(x, ρ) ∩ (R \X) 6= ∅. The set of all boundary points of X is called the boundary of X and is denoted by bd X.

iii) Let A = aff X. x ∈ X is called a relative interior point of X if there exists a ρ > 0 such that Bn(x, ρ) ∩ A ⊆ X. The set of all relative interior points is called the relative interior of X and is denoted by relint X.

iv) Let A = aff X. x ∈ A is called a relative boundary point of X if for all ρ > 0, Bn(x, ρ) ∩ X 6= ∅ and Bn(x, ρ) ∩ (A\X) 6= ∅. The set of all relative boundary points of X is called relative boundary of X and is denoted by relbd X.

n 0.14 Remark. Let X ⊆ R be closed. Then X = relint X ∪ relbd X.

0.15 Theorem. Let K ∈ Cn, x ∈ relint K and y ∈ K. Then (1 − λ)x + λy ∈ relint K for all λ ∈ [0, 1).

0.16 Corollary. Let K ∈ Cn be closed. Let x ∈ relint K and y ∈ aff K \ K. Then the segment conv {x, y} intersects relbd K in precisely one point.

n 0.17 Definition [Polytope and simplex]. Let X ⊂ R of finite cardinality, i.e., #X < ∞. i) conv X is called a (convex) polytope.

n ii) A polytope P ⊂ R of dimension k is called a k-polytope. iii) If X is affinely independent and dim X = k then conv X is called a k- simplex. 4 Some basic and convex facts

n n 0.18 Notation. P = {P ⊂ R : P polytope} denotes the set of all polytopes n in R .

n 0.19 Lemma. Let T = conv {x1,..., xk+1} ⊂ R be a k-simplex, and let P P λi > 0, 1 ≤ i ≤ k + 1, with λi = 1. Then λi xi ∈ relint T .

0.20 Corollary. Let K ∈ Cn, K 6= ∅. Then relint K 6= ∅.

n n 0.21 Theorem. Let P = conv {x1,..., xm} ∈ P . A point x ∈ R belongs to Pm relint P if and only if x admits a representation as x = i=1 λixi with λi > 0, Pm 1 ≤ i ≤ m, and i=1 λi = 1.

0.22 Notation.

n i) For two sets X,Y ⊆ R the vectorial addition

X + Y = {x + y : x ∈ X, y ∈ Y }

is called the Minkowski 1 sum of X and Y . If X is just a singleton, i.e., X = {x}, then we write x + Y instead of {x} + Y .

n ii) For λ ∈ R and X ⊆ R we denote by λ X the set

λ X = {λ x : x ∈ X} .

For instance, Bn(a, ρ) = a + ρ Bn.

1Hermann Minkowski, 1864–1909 Support and separate 5

1 Support and separate

n + 1.1 Notation. Let a ∈ R , a 6= 0, and α ∈ R. The closed halfspaces H (a, α), − n H (a, α) ⊂ R are given by +  n −  n H (a, α) = x ∈ R : ha, xi ≥ α ,H (a, α) = x ∈ R : ha, xi ≤ α . The hyperplane H(a, α) is defined by  n H(a, α) = x ∈ R : ha, xi = α .

n 1.2 Definition [Supporting hyperplane]. Let X ⊂ R . A hyperplane H(a, α) ⊂ n R is called supporting hyperplane of X if: i) H(a, α) ∩ X 6= ∅ and ii) X ⊆ H−(a, α). a is called outer normal vector of X and if, in addition, |a| = 1 then it is called outer unit normal vector of X.

n 1.3 Proposition. Let X ⊂ R and let H(a, α) be a supporting hyperplane of X. Then H(a, α) ∩ conv X = conv H(a, α) ∩ X.

n n 1.4 Remark. Let X ⊂ R be compact and a ∈ R \{0}. Then there exists a supporting hyperplane of X with outer normal vector a.

1.5 Definition [Nearest point map (or metric projection)]. Let K ∈ Cn be n n closed. The map ΦK : R → K, where for x ∈ R the point ΦK (x) ∈ K is  given by |x − ΦK (x)| = min |x − y| : y ∈ K is called the nearest point map (metric projection) with respect to K.

1.6 Remark. The nearest point map is well-defined: Notice that since K is n  closed, for all x ∈ R there exist yx ∈ K such that |x − yx| = min |x − y| : y ∈ K , and yx is uniquely determined. In fact, if there exists y ∈ K, y 6= yx, with |x − y| = |x − yx| then we may assume that x−yx and x−y are linearly independent. Hence

yx + y 1 1 1 1 x − = (x − y ) + (x − y) < |x − y | + |x − y| = |x − y x| . 2 2 x 2 2 x 2 v

Since K is convex, (yx + y)/2 ∈ K which contradicts the minimality of yx.

n n 1.7 Theorem. Let K ∈ C be closed and let x ∈ R \ K. Let a = x − ΦK (x) and α = ha, ΦK (x)i. Then H(a, α) is a supporting hyperplane of K with outer normal vector a.

n n 1.8 Corollary. Let K ∈ C , K 6= R , be closed. Then \ K = H−(a, α), H(a,α) supporting hyperplane of K i.e., K is the intersection of all its “supporting halfspaces”. 6 Support and separate

n n 1.9 Corollary. Let X ⊂ R such that conv X is closed and conv X 6= R . Then \ conv X = H−(a, α), X⊆H−(a,α) i.e., conv X is the intersection of all halfspaces containing X.

1.10 Lemma [Busemann-Feller Lemma]. 2,3 Let K ∈ Cn be closed. Then

|ΦK (x) − ΦK (y)| ≤ |x − y|

n for all x, y ∈ R , i.e., the nearest point map does not increase distances. In particular, it is a continuous map.

1.11 Theorem. Let K ∈ Cn be compact and let ρ > 0 such that K ⊂ n−1 n−1 int (ρBn). The nearest point map on ρ S , i.e., ΦK : ρ S → bd K is surjective.

1.12 Corollary. Let K ∈ Cn be closed and let x ∈ relbd K. Then there exists a supporting hyperplane H(a, α) of K with x ∈ H(a, α).

H−(a, α) H+(a, α)

K2

K1

H(a, α)

Figure 1: A strictly separating hyperplane of two compact convex sets

n 1.13 Theorem [Separation theorem]. Let K1,K2 ∈ C with K1 ∩ K2 = ∅. Then there exists a separating hyperplane H(a, α) of K1 and K2, i.e., K1 ⊆ + − H (a, α) and K2 ⊆ H (a, α). If K1 is closed and K2 is compact, then there exists even a strictly sepa- + rating hyperplane H(a, α) of K1 and K2, i.e., K1 ⊂ int H (a, α) and K2 ⊂ int H−(a, α).

1.14 Definition [Support function, breadth]. Let K ∈ Cn, K 6= ∅. The n function h(K, ·): R → R given by h(K, u) = suphu, xi : x ∈ K is called support function of K. For u ∈ Sn−1 the breadth of K in the direction u is defined by h(K, u) + h(K, −u).

2Herbert Busemann, 1905–1994 3William Feller, 1906–1970 Support and separate 7

1.15 Proposition. Let K ∈ Cn be non-empty and compact. Then

\  n K = x ∈ R : hu, xi ≤ h(K, u) . u∈Sn−1

n 1.16 Definition [Convex function]. Let K ∈ C . A function f : K → R is called convex if

fλx + (1 − λ)y ≤ λf(x) + (1 − λ)f(y), for all x, y ∈ K, λ ∈ (0, 1). f is called strictly convex when the above inequality holds as a strict inequality if x 6= y. If −f is convex then f is called concave.

1.17 Proposition. Let f : K → R be a function differentiable on an open n convex set K ⊂ R . i) f is convex if and only if

f(x) ≥ f(y) + h∇f(y), x − yi , for all x, y ∈ K. (1.17.1)

ii) Let f be twice differentiable. Then f is convex if and only if its Hessian is positive semi-definite for all points in K.

4 n 1.18 Theorem [Jensen’s inequality]. Let K ∈ C and let f : K → R be Pm convex. For all x1,..., xm ∈ K and 0 ≤ λ1, . . . , λm with i=1 λi = 1 it holds

m ! m X X f λixi ≤ λif(xi). i=1 i=1

1.19 Remark. Let f : K → R be defined on a convex set K. The set epi f = { x  ∈ n+1 : x ∈ K, x ≥ f(x)} is called the epigraph of f. Then f is xn+1 R n+1 convex if and only if its epigraph epi f is convex.

n 1.20 Theorem*. Let K ∈ C be open and let f : K → R be convex. Then f is continuous.

1.21 Theorem. Let K ∈ Cn be bounded and let K 6= ∅.

i) h(K, ·) is a convex function.

ii) h(K, ·) is positively homogeneous of degree 1, i.e., h(K, λu) = λh(K, u) n for all λ ≥ 0 and u ∈ R .

n iii) If h : R → R is a function satisfying i) and ii) then there exists a convex n n closed and bounded set K ∈ C such that h(K, u) = h(u) for all u ∈ R .

4Johan Jensen, 1859–1925 8 Support and separate

n 1.22 Definition [Polar set]. Let X ⊆ R .

?  n X = y ∈ R : hx, yi ≤ 1 for all x ∈ X is called the polar set of X.

1.23 Proposition.

i) X? is a convex and closed set and 0 ∈ X?.

? ? ii) If X1 ⊆ X2 then X2 ⊆ X1 . iii) Let M be a regular n × n matrix. Then (MX)? = M −|X?.

n S ? T ? iv) Let Xi ⊆ R , i ∈ I. Then i∈I Xi = i∈I Xi . v) X ⊆ (X?)?.

n ? vi) Let X ⊂ R . Then X = X if and only if X = Bn.

1.24 Proposition.

n i) Let P = conv {x1,..., xm} ⊂ R . Then

?  n P = y ∈ R : hxi, yi ≤ 1, 1 ≤ i ≤ m .

 n n ii) Let P = x ∈ R : hai, xi ≤ 1, 1 ≤ i ≤ m with ai ∈ R . Then

? P = conv {0, a1,..., am}.

1.25 Lemma. Let K ∈ Cn be closed with 0 ∈ K. Then (K?)? = K. Radon, Helly, Caratheodory and (a few) relatives 9

2 Radon, Helly, Caratheodory and (a few) relatives

5 n 2.1 Theorem [Radon]. Let X ⊂ R . If #X ≥ n + 2 then there exist X1,X2 ⊂ X with X1 ∩ X2 = ∅ and conv X1 ∩ conv X2 6= ∅.

6 n 2.2 Theorem [Helly]. Let K1,...,Km ∈ C , m ≥ n + 1, such that for each T (n + 1)-index set I ⊆ {1, . . . , m} we have i∈I Ki 6= ∅. Then all sets Ki have a Tm point in common, i.e., j=1 Ki 6= ∅.

2.3 Remark.

i) Without any further restrictions/assumptions Helly’s theorem is not true 1 for infinitely many convex sets Ki. For instance, let Ki = (0, i ], i ∈ N. ii) Helly’s theorem, however, can be e generalised to any collection of com- pact (bounded and closed) convex sets due to the “finite intersection prop- erty” of compact sets: a family o compact sets has nonempty intersection proviede each of its finite subfamilies has.

n n 2.4 Corollary. Let C ⊂ C be compact. Then there exists t ∈ R with

−C ⊆ t + n C.

n n 2.5 Definition [Centerpoint]. For a finite point set X ⊂ R a point c ∈ R is called centerpoint if every closed halfspace containing c containes at least 1 b n+1 #Xc points of X.

n 2.6 Theorem. Every finite set X ⊂ R has a centerpoint.

7 n 2.7 Theorem [Carath´eodory]. Let X ⊂ R . Then

(n+1 n+1 ) X X conv X = λi xi : λi ≥ 0, λi = 1, xi ∈ X, i = 1, . . . , n + 1 . i=1 i=1

n 2.8 Remark. Let X ⊂ R . Then

(dim X+1 dim X+1 ) X X conv X = λi xi : λi ≥ 0, λi = 1, xi ∈ X . i=1 i=1

As a direct consequence of Carath´eodory’s Theorem 2.7 we get

2.9 Corollary. A polytope is the union of simplices.

2.10 Corollary. The convex hull of a compact set is compact.

5Johann Karl August Radon, 1887–1956 6Eduard Helly, 1884–1943 7Constantin Carath´eodory, 1873 - 1950 10 Radon, Helly, Caratheodory and (a few) relatives

n 2.11 Theorem [(Weak) Fractional Helly theorem]. Let K1,...,Km ∈ C , m ≥ n + 1, and let α ∈ (0, 1] such that for at least α m  of the (n + 1)-index T n+1 sets I ⊆ {1, . . . , m} we have i∈I Ki 6= ∅. Then there exists a point in common α of at least n+1 · m sets Ki.

n 2.12 Theorem* [(strong) Fractional Helly theorem]. Let K1,...,Km ∈ C , m ≥ n + 1, and let α ∈ (0, 1] such that for at least α m  of the (n + 1)-index T n+1 sets I ⊆ {1, . . . , m} we have i∈I Ki 6= ∅. Then there exists a point in common 1/(n+1) of at least (1 − (1 − α) ) · m sets Ki.

2.13 Remark. The (strong and sharp) fractional Helly theorem, which is due to Kalai, gives that (1 − (1 − α)1/(n+1)) · m sets have a point in common. Obviously, for α = 1 we get again the classcial Helly Theorem 2.2.

n 2.14 Theorem [Colorful Carath´eodory theorem]. Let X1,...,Xn+1 ⊂ R such that 0 ∈ conv Xi, 1 ≤ i ≤ n + 1. There exist xi ∈ Xi, 1 ≤ i ≤ n + 1, such that 0 ∈ conv {x1,..., xn+1}.

8 n 2.15 Theorem [Tverberg]. Let X ⊆ R and let k ∈ N≥1. If #X ≥ (k − 1)(n+1)+1, k ∈ N, then there exist k subsets X1,...,Xk ⊂ X with Xi∩Xj = ∅, i 6= j, but conv X1 ∩ conv X2 ∩ · · · ∩ conv Xk 6= ∅.

n 2.16 Theorem*. Let X ⊂ R and let #X = m ≥ n + 1. Then there exists a n m  point y ∈ R contained in at least γn n+1 X-simplices, i.e., simplices of the form conv S, S ⊆ X, #S = n+1. Here γn is a positive constant depending only on the dimension, and X-simplices conv S1, conv S2 are considered different if S1 6= S2.

8Helge Arnulf Tverberg, 1935– The multifaceted world of polytopes 11

3 The multifaceted world of polytopes

3.1 Definition [Polyhedron]. The intersection of finitely many closed halfs- paces is called a polyhedron.

3.2 Theorem [Minkowski, Weyl]. 9,10

i) A bounded polyhedron is a polytope.

ii) A polytope is a bounded polyhedron.

3.3 Notation [V-Polytope, H-Polytope]. A polytope given as the convex hull of finitely many points is called a V-polytope. If it is given as the bounded intersection of finitely many closed halfspaces, then it is called an H-polytope.

3.4 Corollary. Let P ∈ Pn.

m×n m i) Let A ∈ R and t ∈ R . Then AP + t is a polytope.

n ii) Let U ⊂ R be an affine subspace. Then P ∩ U is a polytope.

3.5 Definition [Faces]. Let K ∈ Cn be closed and let H be a supporting hyperplane of K. If j = dim(K ∩ H), then K ∩ H is called a j-face of K. Moreover, K itself is regarded as a (dim K)-face and the empty set ∅ as (−1)- face of K.

3.6 Notation [Vertices, exposed points, edges, facets]. A 0-face of K ∈ Cn, K closed, is called an exposed point or in the case of a polytope a vertex, a 1-face of a polytope is called edge and a (dim K − 1)-faceof K is called facet of K. K itself and the empty set are called improper faces, whereas the remaining faces are called proper faces of K. The set of all vertices of a polytope P is denoted by vert P .

3.7 Remark.

i) Let K ∈ Cn be closed. Every (relative) boundary point of K lies in a suitable j-face, 0 ≤ j ≤ dim K − 1 (cf. Corollary 1.12).

ii) Let K ∈ Cn, dim K = n. Let F be a facet of K and H a supporting hyperplane of K with F = K ∩ H. Then H = aff F .

iii) A point v ∈ K, K ∈ Cn, is called an extreme point of K, if it can not be written as a convex combination of two other points in K.

3.8 Proposition. Every face of a polytope is a polytope, and a polytope has only finitely many faces.

9Hermann Minkowski, 1864–1909 10Hermann Klaus Hugo Weyl, 1885 – 1955 12 The multifaceted world of polytopes

n 3.9 Definition [f-vector]. For P ∈ P let fi(P ) be the number of i-faces of P , −1 ≤ i ≤ dim P . Furthermore, let fi(P ) = 0 for dim P + 1 ≤ i ≤ n. The vector f(P ) with entries fi(P ), −1 ≤ i ≤ n, is called the f-vector of P .

3.10 Remark.

n+1 i) Let Tn be an n-dimensional simplex. Then fi(Tn) = i+1 , i.e., any (i+1) subset of the vertices are the vertices of an i-face.

n Pn n+1 ii) For any n-polytope P ∈ P we have i=−1 fi(P ) ≥ 2 with equality if and only if P is an n-simplex.

3.11 Lemma. Let P ∈ Pn.

i) v ∈ vert P can not be written as a convex combination of two other points of P , i.e., v ∈/ conv P \{v}.

ii) If P = conv W then vert P ⊆ W .

iii) P = conv (vert P ).

3.12 Lemma. Let P ∈ Pn be an n-polytope with 0 ∈ int P . For a proper face F of P let F  = {y ∈ P ? : hx, yi = 1 for all x ∈ F } . Then

i) F  is a face of P ?.

ii) F = (F ).

iii) If G is a face of P and F ⊆ G, then G ⊆ F .

iv) dim F  = n − 1 − dim F .

3.13 Theorem. Let P ∈ Pn be an n-polytope with 0 ∈ int P . Then

? fn−1−i(P ) = fi(P ), −1 ≤ i ≤ n.

n 3.14 Theorem. Let P ∈ P be an n-polytope with facets F1,...,Fm and let H(ai, αi), 1 ≤ i ≤ m, be the supporting hyperplanes of Fi, 1 ≤ i ≤ m. Then

m \ − P = H (ai, αi). i=1

3.15 Theorem. Let P ∈ Pn be an n-polytope.

i) The boundary of P is the union of all its facets. The multifaceted world of polytopes 13

ii) A k-face is the intersection of (at least) (n − k) facets.

iii) An (n − 2)-face is contained in exactly two facets.

iv) If F,G are faces of P with F ⊆ G, then F is a face of G.

v) A face of P is also a face of a facet of P .

3.16 Theorem. Let P ∈ Pn be an n-polytope.

i) Let G be a face of P and let F be a face of G. Then F is a face of P .

ii) Let Fj be a j-face of P and let Fk be a k-face of P with Fj ⊂ Fk. There exist i-faces Fi of P , j < i < k, such that

Fj ⊂ Fj+1 ⊂ · · · ⊂ Fk−1 ⊂ Fk.

3.17 Remark. Let v0 be a vertex of an n-polytope P and let {v1,..., vr} be all adjacent vertices of v0, i.e., conv {v0, vi} is an edge of P . In other words, {v1,..., vr} are the neighbours of v0. Then

i) P ⊂ v0 + pos {v1 − v0,..., vr − v0}.

n ii) Let c ∈ R with hc, v0i ≥ hc, vii, 1 ≤ i ≤ r. Then

max{hc, xi : x ∈ P } = hc, v0i .

3.18 Theorem [Euler-Poincar´eformula]. 11 12 Let P ∈ Pn. Then

n X i (−1) fi(P ) = 0. (3.18.1) i=−1

In particular, in the 3-dimensional case, i.e., dim P = 3, it holds f0 −f1 +f2 = 2.

3.19 Proposition. The Euler-Poincar´eformula is the only linear equation sat- Pn isfied by the f-vector, i.e., let λi ∈ R, such that i=−1 λi fi(P ) = 0 for all n i P ∈ P . Then there exists a constant γ ∈ R, such that λi = γ (−1) .

3.20 Definition [Simple and simplicial polytopes]. Let P ∈ Pn.

i) P is called simplicial if all proper faces are simplices.

ii) P is called simple if every vertex is contained in exactly dim P many facets.

11Leonhard Euler,1707–1783 12Henri Poincar´e, 1854–1912 14 The multifaceted world of polytopes

3.21 Lemma. Let P ∈ Pn be an n-polytope with 0 ∈ int P . The following statements are equivalent: i) P is simplicial.

ii) All facets of P are simplices.

iii) P ? is simple.

iv) Every k-face of P ? is contained in exactly n−k facets for k = 0, . . . , n−1.

3.22 Theorem. Let P ∈ Pn be a simple n-polytope. Then n i) Every vertex is contained in exactly k k-faces of P , k = 0, . . . , n − 1. ii) The intersection of k facets containing a common vertex is an (n−k)-face of P .

iii) Let v1,..., vn be the neighbours of a vertex v0 of P . For each subset of

k neighbours vi1 ,..., vik there exists a unique k-face F of P containing

v0, vi1 ,..., vik . iv) A face of a simple polytope is simple. n−j v) Every j face of P is contained in exactly k−j k faces of P .

3.23 Theorem. Let P ∈ Pn be a simple n-polytope.

i) n f0(P ) = 2 f1(P ). Pn n ii) k=0 fk(P ) ≤ 2 f0(P ).

iii) f0(P ) ≤ 2fdn/2e(P ).

3.24 Corollary. Let P be a simple n-polytope with m facets. Then

n  m  X  m  f (P ) ≤ 2 and f (P ) ≤ 2n+1 . 0 bn/2c k bn/2c k=0 Or equivalently: Let P be a simplicial n-polytope with m vertices. Then

n  m  X  m  f (P ) ≤ 2 and f (P ) ≤ 2n+1 . n−1 bn/2c k bn/2c k=0

3.25 Lemma*. Let P be an n-polytope. i) There exists a simple n-polytope Q with the same number of facets as P and fi(P ) ≤ fi(Q), 0 ≤ i ≤ n − 2. ii) There exists a simplical n-polytope Q? with the same number of vertices ? as P and fi(P ) ≤ fi(Q ), 1 ≤ i ≤ n − 1. The multifaceted world of polytopes 15

3.26 Corollary. Let P be an n-polytope with m facets. Then  m  f (P ) ≤ 2 . 0 bn/2c Or equivalently: Let P be an n-polytope with m vertices. Then  m  f (P ) ≤ 2 . n−1 bn/2c

n 3.27 Definition [Cyclic polytopes]. The curve γ : R → R given by γ(t) = (t, t2, t3, . . . , tn)| is called moment curve. The convex hull of m points on the moment curve is called a cyclic polytope with m vertices and is denoted by C(n, m).

3.28 Proposition. Any n + 1 points on the moment curve are affinely inde- pendent. In particular, cyclic polytopes are simplicial polytopes.

3.29 Theorem* [McMullen’s Upper Bound Theorem, 1971]. 13 Let P be an n-polytope with m vertices. Then

( (n−1)/2 P i+2 m−j j+1 , n odd, f (P ) ≤ f (C(n, m)) = j=0 m−j j+1 i+1−j i i Pn/2 m m−j j  j=1 m−j j i+1−j , n even. In particular, ( m−bn/2c−1 2 bn/2c , n odd, fn−1(P ) ≤ fn−1(C(n, m)) = m−bn/2c m−bn/2c−1 bn/2c + bn/2c−1 , n even.

For fixed n the right hand sides are of order mbn/2c.

3.30 Theorem* [Barnette’s Lower Bound Theorem, 1973]. 14 Let P be a simplicial n-polytope with m vertices. P has at least as many i-faces as the so called stacked polytopes P (n, m) with m vertices for which

( n n+1 m i − i i+1 , 0 ≤ i ≤ n − 2, fi(P (n, m)) = n + 1 + (m − (n + 1))(n − 1), i = n − 1.

P (n, n + 1) is an n-simplex, and for m ≥ n + 2 an m-vertex stacked n-polytope P (n, m) is the convex hull of an (m − 1)-vertex stacked polytope with an addi- tional point that is beyond exactly one facet.

13Peter McMullen, born 1942 14David W. Barnette 16 The multifaceted world of polytopes

n 3.31 Remark. For any n-polytope P ∈ P we have nf0(P ) ≤ 2 f1(P ) with equality iff P simple and n fn−1(P ) ≤ 2 fn−2(P ) with equality iff P simplicial.

15 3.32 Theorem [Steinitz, 1906]. A non-negative integral vector (f0, f1, f2) is the f-vector of a 3-polytope if and only if i) f0 − f1 + f2 = 2, ii) 3 f0 ≤ 2 f1, and iii) 3 f2 ≤ 2f1.

3.33 Conjecture [Kalai, 1989]. 16 Let P ∈ Pn be a 0-symmetric n-polytope. Then n X n fi(P ) ≥ 3 . i=0

Here we have equality, for instance, for the cube Cn and its polar, the cross- ? poyltope Cn, or, more generally, for the class of Hanner-polytopes. In 2007 the conjecture has been verified for all n ≤ 4 (see http://front.math.ucdavis. edu/0708.3661).

3.34 Theorem [Figiel, Lindenstrauss, Milman, 1977]. 171819 Let P ∈ Pn be a 0-symmetric n-polytope, i.e., P = −P . Then 1 ln(f (P )) ln(f (P )) ≥ n. 0 n−1 16

15Ernst Steinitz, 1871 – 1928 16Gil Kalai, born 1955 17Tadeusz Figiel 18Joram Lindenstrauss, 1936–2012 19Vitali Milman, born 1939 The space of compact sets / convex bodies 17

4 The space of compact sets / convex bodies

n n 4.1 Notation. Let M ⊂ R be the set of all non-empty compact sets, and n n n n let K = C ∩ M be the set of all convex bodies in R , i.e., the set of all non-empty convex compact sets.

n n Pm 4.2 Remark. Let Ki ∈ K (or M ) and λi ∈ R, i = 1, . . . , m. Then i=1 λiKi ∈ Kn (or Mn).

4.3 Definition [Outer parallel body, Hausdorff distance]. 20 n i) For M ∈ M and ρ ∈ R≥0, the set M + ρBn is called the outer parallel body of M at distance ρ.

n ii) Let M1, M2 ∈ M . The Hausdorff distance of M1 and M2 is defined as

δ(M1,M2) = min{ρ ≥ 0 : M1 ⊆ M2 + ρBn and M2 ⊆ M1 + ρBn}.

4.4 Remark. The Hausdorff-distance δ(·, ·) defines a metric on the space Mn.

n 4.5 Definition [Convergent sequences]. A sequence of Ki ∈ K , i ∈ N, is n n called convergent (to K ∈ K ) iff there exists a K ∈ K such that limi→∞ δ(Ki,K) = 0; in this case we also write Ki → K or limi→∞ Ki = K.

n T∞ 4.6 Lemma. Let Ki ∈ K , i ∈ N, with Ki+1 ⊆ Ki and let K = i=1 Ki. Then n K ∈ K and Ki → K.

n 21 4.7 Lemma. Let Ki ∈ K , i ∈ N, be a Cauchy -sequence of convex bodies, i.e., ∀ > 0 ∃ m ∈ N such that δ(Ki,Kj) < , ∀i, j ≥ m. Then there exists a n n K ∈ K with Ki → K. In other words, the space K is complete.

n 4.8 Definition [Bounded sequences]. A sequence Ki ∈ K , i ∈ N, is called bounded, if there exists an γ ∈ R>0 such that Ki ⊆ γ Bn for all i ∈ N.

4.9 Theorem [Selection theorem of Blaschke, 1916]. 22 A bounded sequence n of convex bodies Ki ∈ K , i ∈ N, contains a convergent subsequence. 4.10 Remark. All the previous statements, i.e., Lemma 4.6, Lemma 4.7 and Theorem 4.9 are also true for the space Mn equipped with the Hausdorff- distance.

4.11 Theorem. Let K ∈ Kn and ρ > 0. Then there exists a polytope P ∈ Pn such that P ⊆ K ⊆ P + ρBn, in particular we have δ(K,P ) ≤ ρ.

4.12 Corollary. Let K ∈ Kn with 0 ∈ relint K and let λ > 1. Then there exists a P ∈ Pn such that P ⊂ K ⊂ λP .

20Felix Hausdorff, 1868–1942 21Augustin-Louis Cauchy, 1789-1857 22Wilhelm Blaschke, 1885–1962 18 The space of compact sets / convex bodies A glimpse on mixed volumes 19

5 A glimpse on mixed volumes

n 5.1 Definition [Volume, characteristic function]. Let M ⊆ R . The func- n tion χM : R → {0, 1} given by ( 1, x ∈ M, χ (x) = M 0, x 6∈ M, is called characteristic function of M. If M is bounded and χM is Riemann 23-integrable then Z Z

vol (M) = χM (x) dx = 1 dx Rn M is called the (n-dimensional) volume (Jordan 24-measure) of M. M is also called Riemann or Jordan-measurable.

5.2 Remark.

n i) The volume of a (rectangular) box B = [0, α1]×[0, α2]×· · ·×[0, αn] ⊂ R Qn is given by vol (B) = i=1 αi.

n ii) Compact convex sets are Jordan-measurable. If M ⊂ R is Jordan- 1 1 n measurable then vol (M) = limk→∞ kn # M ∩ k Z . n iii) Let M,M1,M2 ⊂ R , M1 ⊆ M2, be Riemann-integrable. Then

n×n a) vol (AM + t) = | det A|vol (M) for any A ∈ R with det A 6= 0, n and t ∈ R . n b) vol (λ M) = |λ| vol (M) for any λ ∈ R.

c) vol (M1) ≤ vol (M2). d) If dim M ≤ n − 1 then vol M = 0.

n 5.3 Notation. Let K ⊂ R be compact and contained in a j-dimensional affine plane A. The j-dimensional volume of K with respect to A is denoted R by vol j(K), i.e., vol j(K) = A χK (x)dx, where here dx is the j-dimensional volume element with respect to the space A.

25 n 5.4 Lemma [Cavalieri’s principle]. Let K ⊂ R be compact and for t ∈ R n let Kt = K ∩ {x ∈ R : xn = t}. Then Z vol (K) = vol n−1(Kt) dt. R

23Bernhard Riemann, 1826–1866 24Camille Jordan, 1838–1922 25Bonaventura Cavalieri, 1598–1647 20 A glimpse on mixed volumes

n 5.5 Lemma [Pyramid, Prism]. Let Q ⊂ R be an (n − 1)-dimensional com- pact convex set. i) Let P = conv {Q, s} be a pyramid, and let h be the distance between the apex s and aff Q. Then h vol (K) = vol (Q). n n−1

ii) Let P = conv {x + Q, y + Q} be a prism (i.e., conv {x, y} is not parallel to aff Q), and let h be the distance between x+aff Q and y +aff Q. Then

vol (P ) = h vol n−1(Q).

5.6 Proposition. i) If int K 6= ∅ then vol (K) > 0.

n ii) Let K1,K2 ⊂ R be compact. Then

vol (K1 ∪ K2) = vol (K1) + vol (K2) − vol (K1 ∩ K2).

iii) Let K,M ∈ Kn with hx, yi = 0 for all x ∈ K, y ∈ M and let k = dim K and m = dim M. Then dim(K + M) = k + m and

vol k+m(K + M) = vol k(K)vol m(M).

iv) Let W be an n-cube of edge length λ. Then vol (W ) = λn and, in partic- ular, vol [−1, 1]n = 2n.

v) Let T =conv {v0, v1,..., vn} be an n-simplex. Then 1 vol (T ) = det(v1 − v0,..., vn − v0) . n!

n 5.7 Lemma. Let P ∈ P with 0 ∈ int P and let F1,...,Fm be the facets of P n with outer normal vectors ui ∈ R , |ui| = 1. Then m 1 X vol (P ) = vol (F ) h(P, u ). n n−1 i i i=1

n 5.8 Corollary. Let P ∈ P , dim P = n, and let F1,...,Fm be the facets of P n with outer normal vectors ui ∈ R , |ui| = 1. Then m X vol n−1(Fi) ui = 0. i=1 A glimpse on mixed volumes 21

n 5.9 Corollary. Let P ∈ P , dim P = n, and let F1,...,Fm be the facets of P n with outer normal vectors ui ∈ R , |ui| = 1. Then

m 1 X vol (P ) = vol (F ) h(P, u ). n n−1 i i i=1

n 5.10 Theorem* [Minkowski]. Let u1,..., um ∈ R , |ui| = 1, such that 0 ∈ int conv {u1,..., um}. Let f1, . . . , fm ∈ R>0 be positive numbers such that Pm i=1 fi ui = 0. Then there exists an unique (up to translations) n-polytope n P ∈ P with m facets F1,...,Fm such that vol n−1(Fi) = fi and u1,..., um are the outer unit normal vectors of the facets.

n 5.11 Definition [Valuations on Kn]. A function f : K → R is called

i) monotonous, if K1 ⊆ K2 implies f(K1) ≤ f(K2),

ii) continuous, if Ki → K implies f(Ki) → f(K),

iii) homogeneous of degree r if for λ > 0 it holds f(λK) = λrf(K),

n iv) translation invariant, if for t ∈ R it holds f(t + K) = f(K),

n n  v) rotation invariant, if for any rotation g : R → R it holds f g(K) = f(K),

vi) rigid motion invariant, if it is translation and rotation invariant,

n n vii) additive, if for K1,K2 ∈ K with K1 ∪ K2 ∈ K it holds

f(K1 ∪ K2) = f(K1) + f(K2) − f(K1 ∩ K2), viii) simple, if f(K) = 0 for dim K ≤ n − 1.

An additive functional is also called a valuation.

n 5.12 Lemma. vol : K → R≥0 is continuous.

n 5.13 Theorem. vol : K → R≥0 is a monotonous, continuous, rigid motion invariant, simple valuation, which is homogeneous of degree n.

5.14 Theorem* [Hadwiger’s characterization of the volume]. 26 Let φ : Kn → R be a continuous, rigid motion invariant and simple valuation. Then there ex- ists a constant c ∈ R such that φ(K) = c vol (K).

5.15 Definition [Homothetic bodies]. Two convex bodies K,L ∈ Kn are called n homothetic iff there exist µ ≥ 0 and t ∈ R such that K = t + µL.

26Hugo Hadwiger, 1908–1981 22 A glimpse on mixed volumes

27 n 5.16 Theorem [Brunn-Minkowski’s Theorem]. Let K,L ⊂ R be com- pact, and let 0 ≤ λ ≤ 1. Then

vol λ K + (1 − λ) L1/n ≥ λ vol (K)1/n + (1 − λ)vol (L)1/n, (5.16.1) i.e., the n-th root of the volume is a concave function. Moreover, if K and L are convex then equality holds if and only if K and L are homothetic or K and L lie in parallel hyperplanes.

5.17 Corollary. Let K ∈ Kn with K = −K. For a k-dimensinal linear sub- n n space Lk ⊂ R , k ∈ {1, . . . , n−1}, and x ∈ R let v(Lk, x) = vol k(K∩(x+Lk)). Then v(Lk, 0) ≥ v(Lk, x) n for all x ∈ R , i.e., the volume maximal section of a o-symmetric convex body with a plane contains the origin.

5.18 Proposition [Multiplicative version of the B-M inequality (5.16.1)]. n Let K,L ⊂ R be compact. Inequality (5.16.1) is equivalent to

vol λ K + (1 − λ) L ≥ vol (K)λ vol (L)1−λ for all λ ∈ [0, 1]. (5.18.1)

n Pr 5.19 Proposition. Let Ki ∈ K , λi ∈ R≥0, 1 ≤ i ≤ r, and let K = i=1 λi Ki. Pr n Then h(K, ·) = i=1 λi h(Ki, ·) and for u ∈ R we have

r  X h i K ∩ H u, h(K, u) = λi Ki ∩ H u, h(Ki, u) . i=1

n 5.20 Theorem [Mixed volumes]. Let r ∈ N, λi ∈ R≥0 and Ki ∈ K , 1 ≤ i ≤ r. There are non-negative coefficients V(Ki1 ,...,Kin ), called mixed volumes, which are symmetric in the indices, such that

r ! r X X vol λi Ki = λi1 · ... · λin V(Ki1 ,...,Kin ), i=1 i1,...,in=1 P  i.e., vol i λi Ki is a homogeneous polynomial of degree n in the variables

λ1, . . . , λr. Moreover, V(Ki1 ,...,Kin ) depends only on the bodies Ki1 ,...,Kin .

n Pr 5.21 Notation. Let K1,...,Kr ∈ K and let ki ∈ {0, . . . , n} with n = i=1 ki.

i) Instead of V(Ki1 ,...,Kin ) we also write V(K1, k1; ... ; Kr, kr), where the

ki are the multiplicities of the body Ki in the sequence (Ki1 ,...,Kin ).

ii) Let n  be the multinomial coefficient, i.e., n  = n! . k1,...,kr k1,...,kr k1!···kr!

27Hermann Brunn, 1862–1939 A glimpse on mixed volumes 23

5.22 Remark. With the notation above we may write r !   X X n k1 kr vol λiKi = λ1 · ... · λr V(K1, k1; ... ; Kr, kr). k1, . . . , kr i=1 k1+···+kr=n In particular, for r = 2 we get n X n vol (λ K + λ K ) = λi λn−iV(K , i; K , n − i). 1 1 2 2 i 1 2 1 2 i=0

n 5.23 Theorem. Let K1,...,Kn ∈ K .

i) V (Km,...,Km) = vol (Km). n×n n ii) Let A ∈ R , det A 6= 0, and let ti ∈ R , 1 ≤ i ≤ r. Then V(t1 + AK1, t2 + AK2,..., tn + AKn) = | det A| V( K1,...,Kn). iii) Mixed volumes are continuous functions on (Kn)n.

iv) Mixed volumes are linear in each argument, i.e., V(λ K+µ L, K2,...,Kn) = n λ V(K,K2,...,Kn) + µV(L, K2,...,Kn) for µ, λ ∈ R≥0, K,L ∈ K . v) Mixed volumes are valuations in each argument, i.e., for let K, L, K ∪ L ∈ Kn it holds

V(K ∪ L, K2,...,Kn) =

V(K,K2 ...,Kn) + V(L, K2,...,Kn) − V(K ∩ L, K2,...,Kn).

n 5.24 Lemma. Let K1,K2 ∈ K and k1 > dim K1. Then

V(K1, k1; K2, n − k1) = 0.

n n−1 5.25 Lemma. Let P ∈ P , dim P ≥ n − 1, and let u1,..., um ∈ S be the unit outer normal vectors of the (n − 1)-faces Fi of P , 1 ≤ i ≤ m. Let n G1,...,Gp be the (n − 2)-faces of P and let K ∈ K . For λ ≥ 0 it holds m p X X  vol (P ) + λ h(K, ui)vol n−1(Fi) + vol Gi + 2λD(K)Bn i=1 i=1 m X ≥ vol (P + λ K) ≥ vol (P ) + λ h(K, ui)vol n−1(Fi), i=1 where D(K) = max|x − y| : x, y ∈ K denotes the of K.

n n−1 5.26 Theorem. Let P ∈ P , dim P ≥ n−1, and let u1,..., um ∈ S be the n unit outer normal vectors of the (n − 1)-faces Fi of P , 1 ≤ i ≤ m. For K ∈ K we have m 1 X V(P, n − 1; K, 1) = h(K, u )vol (F ). n i n−1 i i=1 24 A glimpse on mixed volumes

5.27 Corollary. Let K ∈ Kn and u ∈ Sn−1. Then 2 VK, n − 1; conv {−u, u}, 1 = vol (K|u⊥), n n−1 i where K|u⊥ denotes the orthogonal projection of K onto the hyperplane u⊥ = H(u, 0).

n n 5.28 Theorem*. Let Ki ∈ K , i = 1 . . . , r, K1 ⊆ K1 ∈ K , and ij ∈ {1, . . . , r}, j = 2, . . . , n. Then

V(K1,Ki2 ,...,Kin ) ≤ V(K1,Ki2 ,...,Kin ).

5.29 Definition [Surface area]. Let K ∈ Kn.

F(K) = n V(K, n − 1; Bn, 1)

is called the surface area of K.

n n−1 5.30 Corollary. Let P ∈ P , dim P ≥ n − 1, and let u1,..., um ∈ S be the unit outer normal vectors of the (n − 1)-faces Fi of P , 1 ≤ i ≤ m. Then

m X F(P ) = vol n−1(Fi). i=1

5.31 Proposition.

n i)F: K → R≥0 is a monotonous, continuous, rigid motions invariant valu- ation, which is homogeneous of degree n − 1.

ii)F( K) = 0 if (and only if) dim K ≤ n − 2.

vol (K+λBn)−vol (K) iii)F( K) = limλ→0 λ .

iv)F( Bn) = n vol (Bn).

5.32 Theorem [Minkowski’s inequalities]. Let K,L ∈ Kn with dim K = n. Then

i)V( K, n − 1; L, 1)n ≥ vol (K)n−1vol (L) and equality holds if and only if K and L are homothetic.

ii)V( K, n − 1; L, 1)2 ≥ vol (K) V(K, n − 2; L, 2).

5.33 Corollary [Isoperimetric inequality]. Let K ∈ Kn with dim K = n. Then n n F(K) F(Bn) n n−1 ≥ n−1 = n vol (Bn), vol (K) vol (Bn) and equality holds if and only if K is an n-dimensional ball. INDEX 25

Index C(n, m), 15 Barnette, David, 15 F , 12 Blaschke, Wilhelm, 17 H(a, α),5 boundary,3 H+(a, α),H−(a, α),5 point,3 Sn−1,2 breadth,6 Cn,2 Brunn Kn, 17 Hermann, 22 Pn,4 Brunn-Minkowski n R ,1 Theorem of, 22 aff X,2 Busemann, Herbert,6 Bn,2 Bn(a, ρ),2 Carath´eodory, Constantin,9 bd X,3 Cauchy χM , 19 Augustin-Loius, 17 conv X,2 Cavalieri’s principle, 19 dim X,2 characteristic function, 19 |x|,1 compact set,9 epi f,7 concave function,7 int X,3 cone,2 lin X,2 convergent sequences of convex bodies, H-polytope, 11 17 V-polytope, 11 convex pos X,2 combination,1 relbd X,3 function,7 relint X,3 hull,2 vert K, 11 set,2 vol (K), 19 convex bodies, 17 vol j(K), 19 convexly dependent,1 x|y,1 cyclic polytope, 15 f-vector, 12 diameter, 23 Cavalieri dimension,2 Bonaventura, 19 edge, 11 adjacent vertex, 13 epigraph,7 affine Euclidean combination,1 inner product,1 hull,2 norm,1 subspace,1 space,1 affinely Euler, Leonhard, 13 dependent,1 Euler-Poincar´eformula, 13 independent,1 exposed point, 11 extreme point, 11 ball,2 Barnette’s Lower Bound Theorem, 15 faces, 11 26 INDEX facet, 11 hull,2 family of convex sets,2 subspace,1 family of polytopes,4 linearly Feller, William,6 dependent,1 Figiel, Tadeusz, 16 independent,1 function additive, 21 McMullen’s Upper Bound Theorem, 15 continuous, 21 McMullen, Peter, 15 homogeneous of degree r, 21 metric projection,5 monotonous, 21 Milman, Vitali, 16 rigid motion invariant, 21 Minkowski, 21 rotation invariant, 21 Minkowski sum,4 simple, 21 Minkowski’s inequalities, 24 translation invariant, 21 Minkowski, Hermann,4, 11 mixed volumes, 22 Hadwiger moment curve, 15 Hugo, 21 multinomial coefficient, 22 volume characterization, 21 nearest point map,5 halfspace,5 neighbour, 13 Hausdorff distance, 17 neighbours, 13 Hausdorff, Felix, 17 Helly, Eduard,9 outer normal vector,5 homothetic bodies, 21 outer parallel body, 17 hyperplane,5 outer unit normal vector,5 Separating,6 Supporting,5 Poincar´e,Henri, 13 polar set,8 improper faces, 11 polyhedron, 11 inequality polytope Isoperimetric, 24 k-polytope,3 Minkowski, 24 positive of Jensen,7 combination,1 interior,3 hull,2 point,3 positively dependent,1 prism, 20 Jensen’s inequality,7 proper faces, 11 Jensen, Johan,7 pyramid, 20 Jordan Camille, 19 Radon, Johann,9 Jordan measurable, 19 relative boundary,3 Kalai, Gil, 16 boundary point,3 interior,3 Lemma interior point,3 of Busemann-Feller,6 Riemann Lindenstrauss, Joram, 16 Benhard, 19 linear combination,1 separating hyperplane,6 INDEX 27 simple polytopes, 13 simplex k-simplex,3 simplicial polytopes, 13 stacked polytopes, 15 Steinitz’s theorem, 16 Steinitz, Ernst, 16 strictly convex function,7 support function,6 supporting hyperplane,5 surface area, 24

Theorem of Carath´eodory,9 of Helly,9 of Radon,9 theorem of Brunn-Minkowski, 22 of Separation,6 Tverberg, Helge, 10 unit ball,2 valuation, 21 vertex, 11 volume, 19 j-dimensional, 19 mixed, 22 polytope, 21 prism, 20 pyramid, 20 simplex, 20

Weyl, Hermann, 11