https://hal.archives-ouvertes.fr/hal-01590242Fradelizi, 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 ScientifiqueMadiman, MokshayMokshayMadimanUniversity of Delaware - Department of Mathematical Sciences - University of Delaware [Newark]Marsiglietti, ArnaudArnaudMarsigliettiCALTECH - California Institute of TechnologyZvavitch, ArtemArtemZvavitchDepartment of Mathematical Sciences Kent State University, - Kent State UniversityOn the monotonicity of Minkowski sums towards convexityHAL CCSD2017Sumsets Brunn-Minkowski supermodular Shapley-Folkman theorem convexhull inner radius Hausdorff distance discrepancy[MATH.MATH-FA] Mathematics [math]/Functional Analysis [math.FA]Fradelizi, Matthieu2017-09-19 14:12:382022-09-29 14:21:152017-09-19 15:01:11enPreprints, Working Papers, ...application/pdf1Let us define for a compact set A ⊂ R n the sequence A(k) = a 1 + · · · + a k k : a 1 ,. .. , a k ∈ A = 1 k A + · · · + A k times. By a theorem of Shapley, Folkman and Starr (1969), A(k) approaches the convex hull of A in Hausdorff distance as k goes to ∞. Bobkov, Madiman and Wang (2011) conjectured that Vol n (A(k)) is non-decreasing in k, where Vol n denotes the n-dimensional Lebesgue measure, or in other words, that when one has convergence in the Shapley-Folkman-Starr theorem in terms of a volume deficit, then this convergence is actually monotone. We prove that this conjecture holds true in dimension 1 but fails in dimension n ≥ 12. We also discuss some related inequalities for the volume of the Minkowski sum of compact sets, showing that this is fractionally superadditive but not supermodular in general, but is indeed supermodular when the sets are convex. Then we consider whether one can have monotonicity in the Shapley-Folkman-Starr theorem when measured using alternate measures of non-convexity, including the Hausdorff distance, effective standard deviation or inner radius, and a non-convexity index of Schneider. For these other measures, we present several positive results, including a strong monotonicity of Schneider's index in general dimension, and eventual monotonicity of the Hausdorff distance and effective standard deviation. Along the way, we clarify the interrelationships between these various notions of non-convexity, demonstrate applications of our results to combinatorial discrepancy theory, and suggest some questions worthy of further investigation.