Let $W$ be a set of points in $\mathbb{R}^n$. Let $C$ be the convex hull of the members of $W$. Is there a simple way of demonstrating that for any $x \in C$ and any $y \in \mathbb{R}^n \backslash C$, there must be some $w \in W$ such that $\|w - x\| < \|w - y\|$?
Distance between points in a convex set and outside of a convex set.
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linear-algebra
geometry