Do Minkowski averages get progressively more convex?

Comptes Rendus

Functional analysis/Geometry

Do Minkowski averages get progressively more convex?

Presented by: Gilles Pisier

Matthieu Fradelizi ¹; Mokshay Madiman ²; Arnaud Marsiglietti ³; Artem Zvavitch ⁴

¹ Laboratoire d'analyse et de mathématiques appliquées, UMR 8050, Université Paris-Est Marne-la-Vallée, 5, bd Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée cedex 2, France
² University of Delaware, Department of Mathematical Sciences, 501 Ewing Hall, Newark, DE 19716, USA
³ Institute for Mathematics and its Applications, University of Minnesota, 207 Church Street SE, 434 Lind Hall, Minneapolis, MN 55455, USA
⁴ Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA

Comptes Rendus. Mathématique, Volume 354 (2016) no. 2, pp. 185-189.

Abstract
Résumé

Let us define, for a compact set $A \subset R^{n}$ , the Minkowski averages of A:

A (k) = {\frac{a_{1} + \dots + a_{k}}{k} : a_{1}, \dots, a_{k} \in A} = \frac{1}{k} (\underset{k times}{\underset{︸}{A + \dots + A}}) .

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 monotonicity holds, and for the Hausdorff distance, we establish that the sequence is eventually nonincreasing.

Pour tout ensemble compact $A \subset R^{n}$ , définissons ses moyennes de Minkowski par

A (k) = {\frac{a_{1} + \dots + a_{k}}{k} : a_{1}, \dots, a_{k} \in A} = \frac{1}{k} (\underset{k fois}{\underset{︸}{A + \dots + A}}) .

Nous étudions la monotonie de la convergence de

A (k)

vers l'enveloppe convexe de A, mesurée par la distance de Hausdorff, le déficit volumique et par l'indice de non-convexité de Schneider. Pour le déficit volumique, nous démontrons que la propriété de monotonie n'est pas satisfaite en général, réfutant ainsi une conjecture de Bobkov, Madiman et Wang. Pour l'index de non-convexité de Schneider, nous montrons une propriété renforcée de monotonie, tandis que, pour la distance de Hausdorff, nous établissons que la suite est décroissante à partir d'un certain rang.

Received: 2015-11-06
Accepted: 2015-12-11
Published online: 2016-01-27

DOI: 10.1016/j.crma.2015.12.005

Author's affiliations:

Matthieu Fradelizi ¹; Mokshay Madiman ²; Arnaud Marsiglietti ³; Artem Zvavitch ⁴

¹ Laboratoire d'analyse et de mathématiques appliquées, UMR 8050, Université Paris-Est Marne-la-Vallée, 5, bd Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée cedex 2, France
² University of Delaware, Department of Mathematical Sciences, 501 Ewing Hall, Newark, DE 19716, USA
³ Institute for Mathematics and its Applications, University of Minnesota, 207 Church Street SE, 434 Lind Hall, Minneapolis, MN 55455, USA
⁴ Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA

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     author = {Matthieu Fradelizi and Mokshay Madiman and Arnaud Marsiglietti and Artem Zvavitch},
     title = {Do {Minkowski} averages get progressively more convex?},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {185--189},
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     volume = {354},
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     year = {2016},
     doi = {10.1016/j.crma.2015.12.005},
     language = {en},
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Matthieu Fradelizi; Mokshay Madiman; Arnaud Marsiglietti; Artem Zvavitch. Do Minkowski averages get progressively more convex?. Comptes Rendus. Mathématique, Volume 354 (2016) no. 2, pp. 185-189. doi : 10.1016/j.crma.2015.12.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.12.005/

References
Cited by

[1] S.G. Bobkov; M. Madiman; L. Wang Fractional generalizations of Young and Brunn–Minkowski inequalities (C. Houdré; M. Ledoux; E. Milman; M. Milman, eds.), Concentration, Functional Inequalities and Isoperimetry, Contemp. Math., vol. 545, Amer. Math. Soc., 2011, pp. 35-53

[2] W.R. Emerson; F.P. Greenleaf Asymptotic behavior of products $C^{p} = C + \dots + C$ in locally compact abelian groups, Trans. Amer. Math. Soc., Volume 145 (1969), pp. 171-204

[3] M. Fradelizi, M. Madiman, A. Marsiglietti, A. Zvavitch, On the monotonicity of Minkowski sums towards convexity, Preprint.

[4] K. Gyarmati; M. Matolcsi; I.Z. Ruzsa A superadditivity and submultiplicativity property for cardinalities of sumsets, Combinatorica, Volume 30 (2010) no. 2, pp. 163-174

[5] M. Madiman; P. Tetali Information inequalities for joint distributions, with interpretations and applications, IEEE Trans. Inf. Theory, Volume 56 (2010) no. 6, pp. 2699-2713

[6] R. Schneider A measure of convexity for compact sets, Pac. J. Math., Volume 58 (1975) no. 2, pp. 617-625

[7] R.M. Starr Quasi-equilibria in markets with non-convex preferences, Econometrica, Volume 37 (1969) no. 1, pp. 25-38

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