Volume of the Minkowski sums of star-shaped sets

Fradelizi, Matthieu and Lángi, Zsolt and Zvavitch, Artem (2019) Volume of the Minkowski sums of star-shaped sets. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY. ISSN 0002-9939 (In Press)

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Abstract

For a compact set A⊂Rd and an integer k≥1, let us denote by A[k]={a1+⋯+ak:a1,…,ak∈A}=∑i=1kA the Minkowski sum of k copies of A. A theorem of Shapley, Folkmann and Starr (1969) states that 1kA[k] converges to the convex hull of A in Hausdorff distance as k tends to infinity. Bobkov, Madiman and Wang (2011) conjectured that the volume of 1kA[k] is non-decreasing in k, or in other words, in terms of the volume deficit between the convex hull of A and 1kA[k], this convergence is monotone. It was proved by Fradelizi, Madiman, Marsiglietti and Zvavitch (2016) that this conjecture holds true if d=1 but fails for any d≥12. In this paper we show that the conjecture is true for any star-shaped set A⊂Rd for d=2 and d=3 and also for arbitrary dimensions d≥4 under the condition k≥(d−1)(d−2). In addition, we investigate the conjecture for connected sets and present a counterexample to a generalization of the conjecture to the Minkowski sum of possibly distinct sets in Rd, for any d≥7.

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