Is every compact set contained in a finitely generated convex set?

Question

Let $V$ be a normed vector space. Let $A$ be a convex and (norm) compact subset of $V$. Is it true that there exists a convex set $B$ that is generated by finitely many extreme points, such that int$B \supset A$?

It seems clear that the result is true if $V$ is finite dimensional. For then we can assume $V = \mathbb{R}^n$, and it follows that $A$ is bounded and contained in some $n$-cube which is the desired $B$.

Now, I believe the result is false in general if $V$ is infinite dimensional (this seems to follow from some results that I'm currently looking at), but I cannot prove it.

Answer 1

This cannot work: If $B$ is a convex set generated by finitely many points $x_1\dots x_n$ in $X$, then $B$ is contained in a finite-dimensional subspace. If you add the requirement $int B\ne\emptyset$ then the space $V$ has to be finite-dimensional as well.

Hence the claim fails for compact sets that are not contained in a finite-dimensional subspace.