https://hal.archives-ouvertes.fr/hal-02924430Fradelizi, MatthieuMatthieuFradeliziLAMA - Laboratoire d'Analyse et de Mathématiques Appliquées - UPEM - Université Paris-Est Marne-la-Vallée - Fédération de Recherche Bézout - UPEC UP12 - Université Paris-Est Créteil Val-de-Marne - Paris 12 - CNRS - Centre National de la Recherche ScientifiqueLángi, ZsoltZsoltLángiZvavitch, ArtemArtemZvavitchDepartment of Mathematical Sciences Kent State University, - Kent State UniversityVOLUME OF THE MINKOWSKI SUMS OF STAR-SHAPED SETSHAL CCSD2020[MATH.MATH-FA] Mathematics [math]/Functional Analysis [math.FA]Fradelizi, Matthieu2020-08-28 09:36:502022-09-30 04:10:552020-09-01 10:58:08enPreprints, Working Papers, ...application/pdf1For 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.