Approximation of Convex Sets by Homothetic Copies of Polyhedra

We characterize the closed convex sets $K \subset \mathbb{R}^n$ which satisfy the following approximation condition: for any given point $v \in {\rm rint} K$ and a scalar $\varepsilon > 0$, there is a polyhedron $P \subset \mathbb{R}^n$ such that $v \in {\rm rint} P$ and $P \subset K \subset v + (1 + \varepsilon)(P - v)$.