MathDB
IMC 2020 Problem 3

Source: IMC 2020

July 27, 2020
IMCconvex geometrygeometrycombinatorial geometryadvanced fieldsIMC 2020

Problem Statement

Let d≥2d \ge 2 be an integer. Prove that there exists a constant C(d)C(d) such that the following holds: For any convex polytope K⊂RdK\subset \mathbb{R}^d, which is symmetric about the origin, and any ε∈(0,1)\varepsilon \in (0, 1), there exists a convex polytope L⊂RdL \subset \mathbb{R}^d with at most C(d)ε1−dC(d) \varepsilon^{1-d} vertices such that (1−ε)K⊆L⊆K.(1-\varepsilon)K \subseteq L \subseteq K.
Official definitions: For a real α,\alpha, a set T∈RdT \in \mathbb{R}^d is a convex polytope with at most α\alpha vertices, if TT is a convex hull of a set X∈RdX \in \mathbb{R}^d of at most α\alpha points, i.e. T={∑x∈Xtxx∣tx≥0,∑x∈Xtx=1}.T = \{\sum\limits_{x\in X} t_x x | t_x \ge 0, \sum\limits_{x \in X} t_x = 1\}. Define αK={αx∣x∈K}.\alpha K = \{\alpha x | x \in K\}. A set T∈RdT \in \mathbb{R}^d is symmetric about the origin if (−1)T=T.(-1)T = T.
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