Powers and logarithms of convex bodies

Milman, Vitali; Rotem, Liran

doi:10.1016/j.crma.2017.09.002

Functional analysis/Geometry

Powers and logarithms of convex bodies
[Puissances et logarithmes de corps convexes]

Présenté par : Mikhaël Gromov

Milman, Vitali ¹ ; Rotem, Liran ²

¹ School of Mathematical Science, Tel Aviv University, Tel Aviv 69978, Israel
² School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

Comptes Rendus. Mathématique, Tome 355 (2017) no. 9, pp. 981-986.

Résumé
Abstract

Existe-t-il suffisamment de corps convexes que nous comprenions vraiment ? L'éventail usuel d'exemples est-il assez diversifié pour saisir la notion de convexité ? Dans cette note, nous proposons une augmentation drastique du corpus d'exemples. Plus précisément, nous présentons plusieurs constructions nouvelles de corps convexes : la moyenne géométrique de deux corps convexes, la fonction puissance $K^{α}$ (qui, en général, n'existe que pour $| α | \leq 1$ ), et même le logarithme $\log K$ .

Do we have enough examples of convex bodies that we truly understand? Is out standard set of examples diverse enough to understand convexity? In this note, we will dramatically increase our set of examples. More specifically, we will present several new constructions of convex bodies: the geometric mean of two convex bodies, the power function $K^{α}$ (which in general exists only for $| α | \leq 1$ ), and even the logarithm $\log K$ .

Reçu le : 2017-08-09
Accepté le : 2017-09-11
Publié le : 2017-09-01

DOI : 10.1016/j.crma.2017.09.002

Affiliations des auteurs :

Milman, Vitali ¹ ; Rotem, Liran ²

¹ School of Mathematical Science, Tel Aviv University, Tel Aviv 69978, Israel
² School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

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Milman, Vitali; Rotem, Liran. Powers and logarithms of convex bodies. Comptes Rendus. Mathématique, Tome 355 (2017) no. 9, pp. 981-986. doi : 10.1016/j.crma.2017.09.002. http://www.numdam.org/articles/10.1016/j.crma.2017.09.002/

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^☆ The first named author is supported by ISF grant number 519/17, and the second named author is supported by an AMS–Simons Travel Grant. Both authors are also supported by BSF grant number 2016050. Part of the research was conducted while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, supported by NSF grant DMS-1440140.