SCHWEITZER 2019 Problem 5

Let $S \subset \mathbb{R}^d$ be a convex compact body with nonempty interior. Show that there is an $\alpha > 0$ such that if $S = \cap_{i \in I} H_i$ , where $I$ is an index set and $(H_i)_{i \in I}$ are halfspaces, then for any $P \in \mathbb{R}^d$ , there is an $i \in I$ for which $\mathrm{dist}(P, H_i) \ge \alpha \mathrm{dist}(P, S)$ .