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