Approximation of Convex Sets by Homothetic Copies of Polyhedra

We characterize the closed convex sets

which satisfy the following approximation condition: for any given point

v \in {\rm rint} K v ∈ r i n t K

and a scalar

\varepsilon > 0 ε > 0

, there is a polyhedron

P \subset \mathbb{R}^n P ⊂ R n

such that

v \in {\rm rint} P v ∈ r i n t P

and

P \subset K \subset v + (1 + \varepsilon)(P - v) P ⊂ K ⊂ v + ( 1 + ε ) ( P − v )

.