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Polytopes Course Notes Polytopes Course Notes Carl W. Lee Department of Mathematics University of Kentucky Lexington, KY 40506 [email protected] Fall 2013 i Contents 1 Polytopes 1 1.1 Convex Combinations and V-Polytopes . 1 1.2 Linear Inequalities and H-Polytopes . 8 1.3 H-Polytopes are V-Polytopes . 10 1.4 V-Polytopes are H-Polytopes . 13 2 Theorems of the Alternatives 14 2.1 Systems of Equations . 14 2.2 Fourier-Motzkin Elimination | A Starting Example . 16 2.3 Showing our Example is Feasible . 19 2.4 An Example of an Infeasible System . 20 2.5 Fourier-Motzkin Elimination in General . 22 2.6 More Alternatives . 24 2.7 Exercises: Systems of Linear Inequalities . 25 3 Faces of Polytopes 28 3.1 More on Vertices . 28 3.2 Faces . 30 3.3 Polarity and Duality . 32 4 Some Polytopes 36 4.1 Regular and Semiregular Polytopes . 36 4.2 Polytopes in Combinatorial Optimization . 38 5 Representing Polytopes 42 5.1 Schlegel Diagrams . 42 5.2 Gale Transforms and Diagrams . 42 6 Euler's Relation 44 6.1 Euler's Relation for 3-Polytopes . 44 6.2 Some Consequences of Euler's Relation for 3-Polytopes . 46 6.3 Euler's Relation in Higher Dimensions . 48 6.4 Gram's Theorem . 51 7 The Dehn-Sommerville Equations 52 7.1 3-Polytopes . 52 7.2 Simple Polytopes . 52 ii 7.3 Simplicial Polytopes . 54 d 7.4 The Affine Span of f(Ps ) ............................ 57 7.5 Vertex Figures . 58 7.6 Notes . 60 8 Shellability 61 9 The Upper Bound Theorem 66 10 The Lower Bound Theorem 71 10.1 Stacked Polytopes . 71 10.2 Motion and Rigidity . 72 10.3 Stress . 76 10.4 Infinitesimal Rigidity of Simplicial Polytopes . 77 10.5 Kalai's Proof of the Lower Bound Theorem . 80 10.6 The Stress Polynomial . 82 11 Simplicial Complexes 85 11.1 The Kruskal-Katona Theorem . 85 11.2 Order Ideals of Monomials . 87 11.3 Shellable Simplicial Complexes . 88 12 The Stanley-Reisner Ring 91 12.1 Overview . 91 12.2 Shellable Simplicial Complexes are Cohen-Macaulay . 91 iii 1 Polytopes Two excellent references are [16] and [51]. 1.1 Convex Combinations and V-Polytopes 1 m n Definition 1.1 Let v ; : : : ; v be a finite set of points in R and λ1; : : : ; λm 2 R. Then m X j λjv j=1 1 m is called a linear combination of v ; : : : ; v . If λ1; : : : ; λm ≥ 0 then it is called a nonnegative (or conical) combination. If λ1 + ··· + λm = 1 then it is called an affine combination. If both λ1; : : : ; λm ≥ 0 and λ1 + ··· + λm = 1 then it is called convex combination. Note: We will regard an empty linear or nonnegative combination as equal to the point O, but will not consider empty affine or convex combinations. Exercise 1.2 Give some examples of linear, nonnegative, affine, and convex combinations in R1, R2, and R3. Include diagrams. 2 Definition 1.3 The set fv1; : : : ; vmg ⊂ Rn is linearly independent if the only solution to Pm j j=1 λjv = O is the trivial one: λj = 0 for all j. Otherwise the set is linearly dependent. Exercise 1.4 Prove that the set S = fv1; : : : ; vmg ⊂ Rn is linearly dependent if and only if there exists k such that vk can be written as a linear combination of the elements of S nfvkg. 2 Solution: Assume S is linearly dependent. Then there is a nontrivial solution to Pm j j=1 λjv = O. So there exists k such that λk 6= 0. Then −λ vk = X j vj: j6=k λk Therefore vk can be written as a linear combination of the elements of S n fvkg. Conversely, suppose there is a k such that vk can be written as a linear combination of k k P j Pm j the elements of S n fv g, say v = j6=k λjv . Set λk = −1. Then j=1 λjv = O provides a nontrivial linear combination of the elements of S equaling O. Therefore S is linearly dependent. 1 Definition 1.5 The set fv1; : : : ; vmg ⊂ Rn is affinely independent if the only solution to Pm j Pm j=1 λjv = O, j=1 λj = 0, is the trivial one: λj = 0 for all j. Otherwise the set is affinely dependent. Exercise 1.6 Prove that the set S = fv1; : : : ; vmg ⊂ Rn is affinely dependent if and only if there exists k such that vk can be written as an affine combination of the elements of S n fvkg. 2 Solution: Assume S is affinely dependent. Then there is a nontrivial solution to Pm j Pm j=1 λjv = O, j=1 λj = 0. So there exists k such that λk 6= 0. Then −λ vk = X j vj: j6=k λk Note that P −λj 1 P = (− λj) j6=k λk λk j6=k 1 = λk λk = 1: Therefore vk can be written as an affine combination of the elements of S n fvkg. Conversely, suppose there is a k such that vk can be written as an affine combination k k P j P of the elements of S n fv g, say v = j6=k λjv , where j6=k λj = 1. Set λk = −1. Then Pm j j=1 λjv = O provides a nontrivial affine combination of the elements of S equaling O, Pm since j=1 λj = 0. Exercise 1.7 Prove that the set fv1; : : : ; vmg is affinely independent if and only if the set fv1 − vm; v2 − vm; : : : ; vm−1 − vmg is linearly independent. 2 Solution: Assume that the set fv1; : : : ; vmg is affinely independent. Assume that Pm−1 j m j=1 λj(v − v ) = O: We need to prove that λj = 0, j = 1; : : : ; m − 1. Now Pm−1 j Pm−1 m Pm−1 Pm j Pm j=1 λjv −( j=1 λj)v = O: Set λm = − j=1 λj. Then j=1 λjv = O and j=1 λj = 0. 1 m Because the set fv ; : : : ; v g is affinely independent, we deduce λj = 0, j = 1; : : : ; m. There- fore the set fv1 − vm; : : : ; vm−1 − vmg is linearly independent. Conversely, assume that the set fv1 −vm; : : : ; vm−1 −vmg is linearly independent. Assume Pm j Pm that j=1 λjv = O and j=1 λj = 0. We need to prove that λj = 0, j = 1; : : : ; m. Now Pm−1 Pm−1 j Pm−1 m Pm−1 j m λm = − j=1 λj, so j=1 λjv −( j=1 λj)v = O. This is equivalent to j=1 λj(v −v ) = 1 m m−1 m O. Because the set fv − v ; : : : ; v − v g is linear independent, we deduce λj = 0, Pm−1 1 m j = 1; : : : ; m − 1. But λm = − j=1 λj, so λm = 0 as well. Therefore the set fv ; : : : ; v g is affinely independent. 2 Definition 1.8 A subset S ⊆ Rn is a subspace (respectively, cone, affine set, convex set) if it is closed under all linear (respectively, nonnegative, affine, convex) combinations of its elements. Note: This implies that subspaces and cones must contain the point O. Exercise 1.9 Give some examples of subspaces, cones, affine sets, and convex sets in R1, R2, and R3. 2 Exercise 1.10 Is the empty set a subspace, a cone, an affine set, a convex set? Is Rn a subspace, a cone, an affine set, a convex set? 2 Exercise 1.11 Prove that a nonempty subset S ⊆ Rn is affine if and only if it is a set of the form L + x, where L is a subspace and x 2 Rn. 2 Exercise 1.12 Are the following sets subspaces, cones, affine sets, convex sets? 1. fx 2 Rn : Ax = Og for a given matrix A. Solution: This set is a subspace, hence also a cone, an affine set, and a convex set. 2. fx 2 Rn : Ax ≤ Og for a given matrix A. Solution: This is a cone, hence also convex. But it is not necessarily a subspace or an affine set. 3. fx 2 Rn : Ax = bg for a given matrix A and vector b. Solution: this set is an affine set, hence also convex. But it is not necessarily a subspace or a cone. 4. fx 2 Rn : Ax ≤ bg for a given matrix A and vector b. Solution for this case: This set is convex. By Proposition 1.13 it suffices to show that it is closed under convex combinations of two elements. So assume Av1 ≤ b and 2 Av ≤ b, and λ1; λ2 ≥ 0 such that λ1 + λ2 = 1. We need to verify that Av ≤ b, 1 2 1 2 1 where v = λ1v + λ2v . But Av = A(λ1v + λ2v ) = λ1Av + λ2Av2. Knowing that 1 2 1 2 Av ≤ b, Av ≤ b and λ1; λ2 ≥ 0, we deduce that λ1Av ≤ λ1b and λ2Av ≤ λ2b. Thus 1 λ1Av + λ2Av2 ≤ λ1b + λ2b = (λ1 + λ2)b = b. Therefore Av ≤ b and the set is closed under pairwise convex combinations. 2 Proposition 1.13 A subset S ⊆ Rn is a subspace (respectively, a cone, an affine set, a convex set) if and only it is closed under all linear (respectively, nonnegative, affine, convex) combinations of pairs of elements. 3 Proof. Exercise. 2 Solution for convex combinations: It is immediate that if S is convex then it is closed under convex combinations of pairs of elements. So, conversely, assume that S is closed under convex combinations of pairs of elements. We will prove by induction on m ≥ 1 that 1 m Pm Pm j if v ; : : : ; v 2 S, λ1; : : : ; λm ≥ 0, and j=1 λj = 1, then v 2 S, where v = j=1 λjv .
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