Do Minkowski averages get progressively more convex?

TL;DR

This paper investigates how Minkowski averages of a set in Euclidean space approach convexity, revealing that certain measures of non-convexity do not always improve monotonically, but some do under specific conditions.

Contribution

The paper disproves a conjecture about volume deficit monotonicity and establishes monotonicity results for Schneider's non-convexity index and the Hausdorff distance.

Findings

Volume deficit monotonicity fails in general.

Strong monotonicity holds for Schneider's non-convexity index.

Sequence becomes eventually nonincreasing for Hausdorff distance.

Abstract

Let us define, for a compact set $A \subset \mathbb{R}^n$, the Minkowski averages of $A$: $$ A(k) = \left\{\frac{a_1+\cdots +a_k}{k} : a_1, \ldots, a_k\in A\right\}=\frac{1}{k}\Big(\underset{k\ {\rm times}}{\underbrace{A + \cdots + A}}\Big). $$ We study the monotonicity of the convergence of $A(k)$ towards the convex hull of $A$, when considering the Hausdorff distance, the volume deficit and a non-convexity index of Schneider as measures of convergence. For the volume deficit, we show that monotonicity fails in general, thus disproving a conjecture of Bobkov, Madiman and Wang. For Schneider's non-convexity index, we prove that a strong form of monotonicity holds, and for the Hausdorff distance, we establish that the sequence is eventually nonincreasing.