# VOLUME OF THE MINKOWSKI SUMS OF STAR-SHAPED SETS

Abstract : For a compact set A ⊂ R d and an integer k ≥ 1, let us denote by A[k] = {a 1 + · · · + a k : a 1 ,. .. , a k ∈ A} = k i=1 A the Minkowski sum of k copies of A. A theorem of Shapley, Folkmann and Starr (1969) states that 1 k A[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 1 k A[k] is non-decreasing in k , or in other words, in terms of the volume deficit between the convex hull of A and 1 k A[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 ⊂ R d for arbitrary dimensions d ≥ 1 under the condition k ≥ d − 1. 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 R d , for any d ≥ 7.
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https://hal-upec-upem.archives-ouvertes.fr/hal-02924430
Submitted on : Friday, August 28, 2020 - 9:36:50 AM
Last modification on : Wednesday, September 9, 2020 - 2:34:36 PM

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FLZ-arxiv-1910.06146.pdf
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• HAL Id : hal-02924430, version 1

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Matthieu Fradelizi, Zsolt Lángi, Artem Zvavitch. VOLUME OF THE MINKOWSKI SUMS OF STAR-SHAPED SETS. 2020. ⟨hal-02924430⟩

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