Thrifty approximations of convex bodies by polytopes Alexander - PowerPoint PPT Presentation

Thrifty approximations of convex bodies by polytopes Alexander Barvinok November 28, 2016, ICERM http://www.math.lsa.umich.edu/ ∼ barvinok/papers.html Alexander Barvinok Thrifty approximations of convex bodies by polytopes

The problem Let B ⊂ R d be a convex body containing the origin in its interior. Given τ > 1, we want to find a polytope with as few vertices as possible, such that P ⊂ B ⊂ τ P . 0 P � P B Most of the time, B is symmetric about the origin, so B = − B and τ measures the Banach-Mazur distance. Alexander Barvinok Thrifty approximations of convex bodies by polytopes

The main result Theorem Let k and d be positive integer and let τ > 1 be a real number such that � 1 / 2 � d + k � k � k � � � τ 2 − 1 � τ 2 − 1 τ − + τ + ≥ 6 . k Then for any symmetric convex body B ⊂ R d there is a symmetric polytope P ⊂ R d with � d + k � N ≤ 8 k vertices such that P ⊂ B ⊂ τ P . Alexander Barvinok Thrifty approximations of convex bodies by polytopes

Fine and coarse approximations Varying k , we get various asymptotic regimes. We will consider two: • τ = 1 + ǫ , ǫ > 0 is small, N is large and k ∼ d √ ǫ ln 1 ǫ . √ • N is polynomial in d , τ ∼ d and k is fixed. Alexander Barvinok Thrifty approximations of convex bodies by polytopes

Fine approximations Corollary For any e √ ≈ 0 . 48 γ > 4 2 there exists ǫ = ǫ 0 ( γ ) > 0 such that for any 0 < ǫ < ǫ 0 and for any symmetric convex body B ⊂ R d there is a symmetric polytope P ⊂ R d with � γ � d √ ǫ ln 1 N ≤ ǫ vertices such that P ⊂ B ⊂ (1 + ǫ ) P . Alexander Barvinok Thrifty approximations of convex bodies by polytopes

Fine approximations Compare with: The “volumetric bound” (Kolmogorov and Tikhomirov 1959?) � γ � d N ≤ ǫ Throw as many points as possible so that the distance between any two (in the � · � B norm) is at least ǫ . Alexander Barvinok Thrifty approximations of convex bodies by polytopes

Fine approximations Compare with: The C 2 -smooth boundary (Gruber 1993): � γ � ( d − 1) / 2 N ≤ for all 0 < ǫ < ǫ 0 ( B ) . ǫ p � N ≈ 1 − π 2 p = cos α = cos π 2 N 2 . Alexander Barvinok Thrifty approximations of convex bodies by polytopes

Coarse approximations Corollary For any 0 < ǫ < 1 , for any d ≥ d 0 ( ǫ ) , for any symmetric convex body B ⊂ R d there is a symmetric polytope P ⊂ R d with N ≤ d 1 /ǫ vertices such that √ P ⊂ B ⊂ ( ǫ d ) P . Alexander Barvinok Thrifty approximations of convex bodies by polytopes

Intermediate regimes � d d τ ≤ γ ln N ln for an absolute constant γ > 0 ln N (suggested to the author in this form by A. Litvak, M. Rudelson and N. Tomczak-Jaegermann, 2012). Alexander Barvinok Thrifty approximations of convex bodies by polytopes

Ideas of the proof: the minimum volume ellipsoid Lemma Let C ⊂ R d be a compact set which spans R d and let E ⊂ R d be the (necessarily unique) ellipsoid of the smallest volume among all ellipsoids centered at the origin and containing C. Suppose that E is the unit ball. Then there exist points x 1 , . . . , x n ∈ C ∩ ∂ E and positive real α 1 , . . . , α n such that n α i � x i , y � 2 = � y � 2 � y ∈ R d . for all i =1 Necessarily, n � α i = d . i =1 This is F. John Theorem (1948). Alexander Barvinok Thrifty approximations of convex bodies by polytopes

Ideas of the proof: the minimum volume ellipsoid E C This produces a set X ⊂ C of n ≤ d ( d + 1) + 1 2 points such that √ max x ∈ X | ℓ ( x ) | ≤ max x ∈ C | ℓ ( x ) | ≤ d max x ∈ X | ℓ ( x ) | for any linear function ℓ : R d − → R . Alexander Barvinok Thrifty approximations of convex bodies by polytopes

Ideas of the proof: sparsification Lemma Let γ > 1 be a real number and let x 1 , . . . , x n be vectors in R d such that n � x i , y � 2 = � y � 2 � y ∈ R d . for all i =1 Then there is a subset J ⊂ { 1 , . . . , n } with | J | ≤ γ d and β j > 0 for j ∈ J such that � γ + 1 + 2 √ γ � � y � 2 ≤ β j � x j , y � 2 ≤ � � y � 2 y ∈ R d . for all γ + 1 − 2 √ γ j ∈ J This is Batson-Spielman-Srivastava Theorem (2008). Alexander Barvinok Thrifty approximations of convex bodies by polytopes

Ideas of the proof: sparsification Given a compact C ⊂ R d , this produces a set X ⊂ C of n ≤ 4 d points such that √ max x ∈ X | ℓ ( x ) | ≤ max x ∈ C | ℓ ( x ) | ≤ 3 d max x ∈ X | ℓ ( x ) | for any linear function ℓ : R d − → R . Alexander Barvinok Thrifty approximations of convex bodies by polytopes

Ideas of the proof: tensorization Let us denote V = R d and let us consider the space W = R ⊕ V ⊕ V ⊗ 2 ⊕ · · · ⊕ V ⊗ k . Let us define a continuous map φ : V − → W by φ ( x ) = 1 ⊕ x ⊕ x ⊗ 2 ⊕ · · · ⊕ x ⊗ k for x ∈ V . We consider the compact set � � C = φ ( x ) : x ∈ B , C ⊂ W . Note that C lies in the symmetric part of W , so � d + 1 � � d + k − 1 � � d + k � dim span( C ) ≤ 1 + d + + . . . + = . 2 k k Alexander Barvinok Thrifty approximations of convex bodies by polytopes

Ideas of the proof: tensorization Pick a set X ⊂ B of � d + k � N ≤ 4 k points such that for any linear function L : W − → R , we have � 1 / 2 � d + k x ∈ X |L ( φ ( x )) | ≤ max max x ∈ B |L ( φ ( x )) | ≤ 3 max x ∈ X |L ( φ ( x )) | . k Alexander Barvinok Thrifty approximations of convex bodies by polytopes

Ideas of the proof: tensorization C B E Define P = conv ( X ∪ − X ) . Alexander Barvinok Thrifty approximations of convex bodies by polytopes

Ideas of the proof: Chebyshev polynomials Recall that V = R d , W = R ⊕ V ⊕ V ⊗ 2 ⊕ · · · ⊕ V ⊗ k and φ : V − → W is defined by φ ( x ) = 1 ⊕ x ⊕ x ⊗ 2 ⊕ · · · ⊕ x ⊗ k for x ∈ V . If L : W − → R is a linear function then L ( φ ( x )) is a polynomial of degree k of x . Suppose that ℓ : R d − → R is linear such that | ℓ ( x ) | ≤ 1 for all x ∈ X . To show that | ℓ ( x ) | ≤ τ for all x ∈ B we would like to construct a polynomial p of degree k such that | p ( t ) | ≤ 1 if | t | ≤ 1 and | p ( τ ) | is the largest possible . Alexander Barvinok Thrifty approximations of convex bodies by polytopes

Ideas of the proof: Chebyshev polynomials Define T k ( t ) = cos ( k arccos t ) provided − 1 ≤ t ≤ 1 T k ( t ) = 1 + 1 � k � k � � � t 2 − 1 � t 2 − 1 t − t + provided | t | > 1 . 2 2 Alexander Barvinok Thrifty approximations of convex bodies by polytopes

Ideas of the proof: Chebyshev polynomials Writing k k � � a i t i , a i ℓ ⊗ i . T k = define L = i =0 i =0 If ℓ ( x ) > τ for some x ∈ B , then for that L we get a contradiction with � 1 / 2 � d + k x ∈ X |L ( φ ( x )) | ≤ max max x ∈ B |L ( φ ( x )) | ≤ 3 max x ∈ X |L ( φ ( x )) | . k Alexander Barvinok Thrifty approximations of convex bodies by polytopes