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arXiv · 1512.03718

Do Minkowski averages get progressively more convex?

Matthieu Fradelizi·Mokshay Madiman·Arnaud Marsiglietti·Artem Zvavitch
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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.

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Matthieu Fradelizi, Mokshay Madiman, Arnaud Marsiglietti, Artem Zvavitch. 2016-02-06. Do Minkowski averages get progressively more convex?. https://doi.org/10.1016/j.crma.2015.12.005

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