MathDB

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)

Problem Statement