Closed bounded convex set in $\mathbb{R}^n$

Question

Let the set $B \subset \mathbb{R}^n$ be convex, bounded and closed. We want to show that set $B$ is equal to convex hull of its boundary?

Answer 1

By Krein-Milman, $$ B = \overline{\mathrm{conv}(\mathrm{ext}(B))} \subseteq \overline{\mathrm{conv}(\partial(B))} \subseteq \overline{\mathrm{conv}(B)} = \overline{B} = B . $$