Discrete Analytic Convex Geometry Introduction

Martin Henk Otto-von-Guericke-Universit¨atMagdeburg Winter term 2012/13 webpage CONTENTS i

Contents

Preface ii

0 Some basic and convex facts1

1 Support and separate5

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

Index 11 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 Algebraic Geometry” – Peter M. Gruber, ”Convex and Discrete Geometry” – 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) dependent 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 convex set]. 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 convex hull, dimension]. 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 point 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 Minkowski1 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)

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.

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 .

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. 8 Support and separate

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

4 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= ∅.

5 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 easily generalised to inﬁnitely many compact (bounded and closed) convex sets.

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.

6 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.

n 2.11 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.

4Johann Karl August Radon, 1887–1956 5Eduard Helly, 1884–1943 6Constantin Carath´eodory, 1873 - 1950 10 Radon, Helly, Caratheodory and (a few) relatives

n 2.12 Theorem [Colorful Carath´eodory theorem]. Let X1,...,Xn+1 ⊂ R be ﬁnite point sets such that 0 ∈ conv Xi, 1 ≤ i ≤ n + 1. There exist xi ∈ Xi, 1 ≤ i ≤ n + 1, such that 0 ∈ conv {x1,..., xn+1}.

7 n 2.13 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 2.14 Theorem*. Let X ⊂ R and let #X = m ≥ n + 1. Then there exists n m a 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.

7Helge Arnulf Tverberg, 1935– INDEX 11

Index 0/1-polytope, 41 Busemann, Herbert, 10 C(n, m), 35 F , 25 Carathéodory, Constantin, 20 H(a, α),9 Combinatorial diameter, 39 H+(a, α),H−(a, α),9 compact set, 19 Sn−1,2 concave function, 13 Cn,2 cone,2 Pn,5 convex n combination,1 R ,1 aff X,3 function, 13 hull,3 Bn,2 set,2 Bn(a, ρ),2 bd X,4 convexly dependent,1 conv X,3 cyclic polytope, 35 dim X,3 Dehn-Sommerville equations, 36 |x|,1 dimension,3 epi f, 14 int X,4 edge, 24 lin X,3 epigraph, 14 H-polytope, 23 Euclidean V-polytope, 23 inner product,1 pos X,3 norm,1 relbd X,4 space,1 relint X,4 Euler-Poincaréformula, 29 vert P , 24 x|y,1 faces, 24 f-vector, 24 facet, 24 h-vector, 37 family of convex sets,2 family of polytopes,5 adjacent vertex, 28 Farkas, Gyula, 17 affine Feller, William, 10 combination,1 function hull,3 gauge, 18 subspace,2 gauge function, 18 affine isomorphic, 41 Graph, 39 affinely dependent,1 halfspace,9 independent,1 Helly, Eduard, 19 Hirsch conjecture, 40 ball,2 hyperplane,9 Barnette’s Lower Bound Theorem, 36 Separating, 12 boundary,4 Supporting,9 point,4 breadth, 13 improper faces, 24 12 INDEX inequality interior point,4 of Jensen, 14 interior,4 separating hyperplane, 12 point,4 simple polytopes, 30 simplex Jensen’s inequality, 14 k-simplex,5 simplicial polytopes, 30 Kalai, 39 stacked polytopes, 36 Steinitz, 37 Lemma strictly convex function, 13 of Busemann-Feller, 10 support function, 13 linear supporting hyperplane,9 combination,1 hull,3 Theorem subspace,2 of Carathéodory, 20 linearly of Helly, 19 dependent,1 of Radon, 19 independent,1 theorem of Separation, 12 McMullen’s g-theorem, 37 Tverberg, Helge, 22 McMullen’s Upper Bound Theorem, 36 metric projection,9 unit ball,2 Minkowski sum,6 unit cube, 41 Minkowski, Hermann,6, 23 moment curve, 35 vector h, 37 nearest point map,9 vertex, 24 neighbour, 28 Weyl, Hermann, 23 outer normal vector,9 outer unit normal vector,9 polar set, 15 polyhedron, 23 Polytope Hanner, 43 polytope k-polytope,5 positive combination,1 hull,3 positively dependent,1 proper faces, 24

Radon, Johann, 19 relative boundary,4 boundary point,4 interior,4