Discrete Analytic Convex Introduction

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

Contents

Preface ii

0 Some basic and convex facts1

Index5 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 = (x1, . . . , xn)| : xi ∈ R denotes the n-dimensional Eu- Pn clidean 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 Remark. 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 , ]. Let X ⊆ n 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 : m ∈ N, xi ∈ X . Pm Pm iii) aff X = i=1 λixi : m ∈ N, xi ∈ X, i=1 λi = 1 . Pm iv) pos X = i=1 λixi : m ∈ N, xi ∈ X, λi ≥ 0 .

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 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} . INDEX 5

Index Sn−1,2 interior,3 Cn,2 point,3 Pn,4 n linear R ,1 aff X,2 combination,1 hull,2 Bn,2 subspace,1 Bn(a, ρ),2 bd X,3 linearly conv X,2 dependent,1 dim X,2 independent,1 |x|,1 Minkowski sum,4 int X,3 lin X,2 polytope pos X,2 k-polytope,3 relbd X,3 positive relint X,3 combination,1 x|y,1 hull,2 positively dependent,1 affine combination,1 relative hull,2 boundary,3 subspace,1 boundary point,3 affinely interior,3 dependent,1 interior point,3 independent,1 simplex ball,2 k-simplex,3 boundary,3 point,3 unit ball,2 cone,2 convex combination,1 hull,2 set,2 convexly dependent,1 dimension,2

Euclidean inner product,1 norm,1 space,1 family of convex sets,2 family of polytopes,4