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On the convergence of Minkowski sums to the convex hull Fradelizi, Matthieu


Let us define, for a compact set $A \subset \R^n$, the Minkowski averages of $A$: $$ A(k) = \left\{\frac{a_1+\cdots +a_k}{k} : a_1, \ldots, a_k\in A\right\}=\frac{1}{k}\Big(\underset{k\ {\rm times}}{\underbrace{A + \cdots + A}}\Big). $$ I shall show some monotonicity properties of $A(k)$ towards convexity when considering, the Hausdorff distance, the volume deficit and a non-convexity index of Schneider. For the volume deficit, the monotonicity holds in dimension 1 but fails for $n\ge 12$, thus disproving a conjecture of Bobkov, Madiman and Wang. For Schneider's non-convexity index, a strong form of monotonicity holds. And for the Hausdorff distance, some monotonicity holds for $k$ large enough, depending of the ambient dimension and not on the set $A$. Based on a work in collaboration with Mokshay Madiman, Arnaud Marsiglietti and Artem Zvavitch.

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