Finite convexification by Minkowski summation

Abstract

Let $A\subset\mathbb{R}^n$ be a nonempty compact set and let $K\subset\mathbb{R}^n$ be a nonempty compact convex set. We study the problem of when there exists $λ>0$ such that the Minkowski sum $A+λK$ is convex. First, we show that this finite convexification is equivalent to a uniform covering condition on the exposed sets of $A$ and $K$. We then show that another characterization is possible in terms of the covering radii of the iterated exposed sets after translating each exposed face of $K$ so that its relative centroid is $0$. In the case of convex polytopes, we show that it suffices to check the exposed sets in the first level of the iteration. We also study several special cases, such as strictly convex convexifiers, lower-dimensional convexifiers, strongly isomorphic polytopes, and the planar case.