Do Minkowski averages get progressively more convex?

Abstract

Let us define, for a compact set $A\subset {\mathbb{R}}^n$, the Minkowski averages of $A$: $$A(k)=\left\{\frac{a_1+\cdots +a_k}{k}:a_1,\dots ,a_k\in A\right\}=\frac{1}{k}\left(\underset{k\mathrm{times}}{\underbrace{A+\cdots +A}}\right).$$ 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.

6 pages, including figures. Contains announcement of results that will be part of a more comprehensive, forthcoming paper. Version 2 corrects a typo

References in corpus (1)

• The convexification effect of Minkowski summation

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