Condition for a compact, convex subset of $\mathbb{R}^n$ to have a nonempty interior?

Question

For $A\subset \mathbb{R}^n$ let $int(A):=\{x \in A: B_\epsilon(x)\cap \mathbb{R}^n\subset A$ for some $\epsilon>0\}$, where $B_\epsilon(x)$ is the Euclidean open ball around $x$.

Is there a known sufficient and/or necessary condition for such $A$ that is also compact and convex to have $int(A)\neq\emptyset$?