On the Closedness of the Convex Hull in a Locally Convex Space

The question of the closedness of the convex hull of the union of a closed convex set and a compact convex set in a locally convex space does not appear to be widely known. We show here that the answer is affirmative if and only if the closed convex set is bounded. The result is first proven for convex compact sets ”of finite type” (polytopes) using an induction argument. It is then extended to arbitrary convex compact sets using the fact that such subsets in locally convex spaces admit arbitrarily small continuous displacements into polytopes.