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Convex Geometry Introduction Convex Geometry 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 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 2 R denotes the n-dimensional Pn Euclidean space equipped with the Euclidean inner product hx; yi = i=1 xi yi, n p x; y 2 R , and the Euclidean norm jxj = hx; xi. 0.2 Definition [Linear, affine, positive and convex combination]. Let m 2 n N and let xi 2 R , λi 2 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 2 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 2 X, m 2 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 2 R are called linearly (affinely) dependent, if one of the xi is linearly (affinely) de- pendent of fx1;:::; xmgnfxig. Otherwise x1;:::; xm are called linearly (affinely) independent. n 0.4 Proposition. Let x1;:::; xm 2 R . x1 xm n+1 i) x1;:::; xm are affinely dependent if and only if 1 ;:::; 1 2 R are linearly dependent. ii) x1;:::; xm are affinely dependent if and only if there exist µi 2 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 2 R which are linearly dependent of X, n ii) affine subspace (set) if it contains all x 2 R which are affinely dependent of X, 2 Some basic and convex facts n iii) (convex) cone if it contains all x 2 R which are positively dependent of X, n iv) convex set if it contains all x 2 R which are convexly dependent of X. n n 0.6 Notation. C = fK ⊆ R : K convexg 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 2 K; for all x; y 2 K and 0 ≤ λ ≤ 1: n 0.8 Example. The closed n-dimensional ball Bn(a; ρ) = x 2 R : jx − aj ≤ ρ with centre a and radius ρ > 0 is convex. The boundary of Bn(a; ρ), i.e., n x 2 R : jx − aj = 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 2 C , i 2 I. Then i2I Ki 2 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 2 N; xi 2 X; λi ≥ 0; λi = 1 : i=1 i=1 0.12 Remark. i) conv fx; yg = λ x + (1 − λ) y : λ 2 [0; 1] . Pm ii) lin X = i=1 λixi : λi 2 R; xi 2 X; m 2 N . Pm Pm iii) aff X = i=1 λixi : λi 2 R; i=1 λi = 1; xi 2 X; m 2 N . Pm iv) pos X = i=1 λixi : λi 2 R; λi ≥ 0; xi 2 X; m 2 N . 0.13 Definition [(Relative) interior point and (relative) boundary point]. n Let X ⊆ R . i) x 2 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 2 R is called boundary point of X if for all ρ > 0, Bn(x; ρ) \ X 6= ; n and Bn(x; ρ) \ (R nX) 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 2 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 2 A is called a relative boundary point of X if for all ρ > 0, Bn(x; ρ) \ X 6= ; and Bn(x; ρ) \ (AnX) 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 2 Cn, x 2 relint K and y 2 K. Then (1 − λ)x + λy 2 relint K for all λ 2 [0; 1). 0.16 Corollary. Let K 2 Cn be closed. Let x 2 relint K and y 2 aff K n K. Then the segment conv fx; yg intersects relbd K in precisely one point. n 0.17 Definition [Polytope and simplex]. Let X ⊂ R of finite cardinality, i.e., #X < 1. 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 = fP ⊂ R : P polytopeg denotes the set of all polytopes n in R . n 0.19 Lemma. Let T = conv fx1;:::; xk+1g ⊂ R be a k-simplex, and let P P λi > 0, 1 ≤ i ≤ k + 1, with λi = 1. Then λi xi 2 relint T . 0.20 Corollary. Let K 2 Cn, K 6= ;. Then relint K 6= ;. n n 0.21 Theorem. Let P = conv fx1;:::; xmg 2 P . A point x 2 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 = fx + y : x 2 X; y 2 Y g is called the Minkowski 1 sum of X and Y . If X is just a singleton, i.e., X = fxg, then we write x + Y instead of fxg + Y . n ii) For λ 2 R and X ⊆ R we denote by λ X the set λ X = fλ x : x 2 Xg : 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 2 R , a 6= 0, and α 2 R. The closed halfspaces H (a; α), − n H (a; α) ⊂ R are given by + n − n H (a; α) = x 2 R : ha; xi ≥ α ;H (a; α) = x 2 R : ha; xi ≤ α : The hyperplane H(a; α) is defined by n H(a; α) = x 2 R : ha; xi = α : n 1.2 Definition [Supporting hyperplane].
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