IMC 2020 Problem 3

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

Official definitions: For a real $\alpha,$ a set $T \in \mathbb{R}^d$ is a convex polytope with at most $\alpha$ vertices, if $T$ is a convex hull of a set $X \in \mathbb{R}^d$ of at most $\alpha$ points, i.e. $T = \{\sum\limits_{x\in X} t_x x | t_x \ge 0, \sum\limits_{x \in X} t_x = 1\}.$ Define $\alpha K = \{\alpha x | x \in K\}.$ A set $T \in \mathbb{R}^d$ is symmetric about the origin if $(-1)T = T.$