Convex hull of convex set boundary

Question

A is a closed convex set with non-empty interior. Does A must equal to the convex hull of its boundary?

I know this is false when A is half space. But what about other sets?

Answer 1

Let $A\subseteq \mathbb{R}^d$ be a convex and closed set with non-empty interior. Then $A$ does not coincide with the convex hull of its boundary if and only if it is either a half-space or $\mathbb{R}^d$ itself. Source: Lemma 1.4.1 in R. Schneider: Convex Bodies: The Brunn--Minkowski Theory, Cambridge University Press, Cambridge (1993). It is a consequence of the separation theorem.