# International Mathematics Competition for University Students 2020

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IMC 2020

## IMC2020: Day 1, Problem 3

Problem 3. Let $\displaystyle d\ge 2$ be an integer. Prove that there exists a constant $\displaystyle C(d)$ such that the following holds: For any convex polytope $\displaystyle K\subset \mathbb{R}^d$, which is symmetric about the origin, and any $\displaystyle \varepsilon\in (0,1)$, there exists a convex polytope $\displaystyle L\subset \mathbb R^d$ with at most $\displaystyle C(d)\varepsilon^{1-d}$ vertices such that

$\displaystyle (1-\varepsilon)K\subseteq L \subseteq K.$

(For a real $\displaystyle \alpha$, a set $\displaystyle T\subset \mathbb{R}^d$ with nonempty interior is a convex polytope with at most $\displaystyle \alpha$ vertices, if $\displaystyle T$ is a convex hull of a set $\displaystyle X\subset\mathbb R^d$ of at most $\displaystyle \alpha$ points, i.e., $\displaystyle T=\{\sum_{x\in X} t_xx\ |\ t_x\ge 0, \sum_{x\in X} t_x = 1\}$. For a real $\displaystyle \lambda$, put $\displaystyle \lambda K=\{\lambda x\ |\ x\in K\}$. A set $\displaystyle T\subset \mathbb{R}^d$ is symmetric about the origin if $\displaystyle (-1) T = T$.)

Fedor Petrov, St. Petersburg State University

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