# Convex hulls of compact sets

Let $A$ be a compact set in a separable Hilbert space $H$, and let $\bar A$ denote its convex hull. Is $\bar A$ compact?

• Not necessarily. Proof by google, counterexample here. – Francois Ziegler Jan 31 '14 at 19:45
• Via Theorem 5.35 on that page, the answer is "yes" if one replaces "convex hull" with "closed convex hull". – Tom LaGatta Jan 31 '14 at 22:42
• (note: that theorem requires that $H$ be locally convex and completely metrizable, which is satisfied for a Hilbert space $H$. The assumption of separability is not necessary) – Tom LaGatta Feb 1 '14 at 0:52
• mathoverflow.net/questions/6627/convex-hull-in-cat0 this related question is purportedly open – Paul Fabel Feb 9 '14 at 17:47
• In any locally convex space $E$, the closed convex hull of a precompact set $X$ is precompact (see Schaefer's Top. Vect. Sp., Chapter II, Section 4.3). It follows that if $E$ is quasi-complete (every closed bounded set is complete - automatically true if $E$ is complete), then the closed convex hull of $X$ is compact, being precompact and complete. – Robert Furber Apr 18 '19 at 2:48