Abstract of a paper by Matthieu Fradelizi, Mokshay Madiman, Arnaud Marsiglietti, and Artem Zvavitch
This is an announcement for the paper "Do Minkowski averages get progressively more convex?" by Matthieu Fradelizi, Mokshay Madiman, Arnaud Marsiglietti, and Artem Zvavitch. Abstract: Let us define, for a compact set $A \subset \mathbb{R}^n$, the Minkowski averages of $A$: $$ A(k) =3D \left\{\frac{a_1+\cdots +a_k}{k} : a_1, \ldo= ts, a_k\in A\right\}=3D\frac{1}{k}\Big(\underset{k\ {\rm times}}{\underbrace{= A + \cdots + A}}\Big). $$ 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 monotonic= ity holds, and for the Hausdorff distance, we establish that the sequence is eventually nonincreasing. Archive classification: math.FA math.OC Remarks: 6 pages, including figures. Contains announcement of results th= The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1512.03718 or http://arXiv.org/abs/1512.03718
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alspach@math.okstate.edu