Local minimality of the volume-product at the simplex

Jaegil Kim; Shlomo Reisner

doi:10.1112/S0025579310001555

Local minimality of the volume-product at the simplex

Jaegil Kim, Shlomo Reisner

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

It is proved that the simplex is a strict local minimum for the volume product, P(K)=min _z∈int(K)|K||K^z|, in the Banach-Mazur space of n-dimensional (classes of) convex bodies. Linear local stability in the neighborhood of the simplex is proved as well. The proof consists of an extension to the non-symmetric setting of methods that were recently introduced by Nazarov, Petrov, Ryabogin and Zvavitch, as well as proving results of independent interest concerning stability of square order of volumes of polars of non-symmetric convex bodies.

Original language	English
Pages (from-to)	121-134
Number of pages	14
Journal	Mathematika
Volume	57
Issue number	1
DOIs	https://doi.org/10.1112/S0025579310001555
State	Published - Jan 2011

Bibliographical note

Funding Information:
Acknowledgements. It is with great pleasure that we thank Karoly Böröczky, Mathieu Meyer, Dmitry Ryabogin and Artem Zvavitch for very helpful advice during the preparation of this paper. The first author was supported in part by U.S. National Science Foundation grant DMS-0652684. The second author was supported in part by the France–Israel Research Network Program in Mathematics contract #3-4301.

ASJC Scopus subject areas

General Mathematics

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title = "Local minimality of the volume-product at the simplex",

abstract = "It is proved that the simplex is a strict local minimum for the volume product, P(K)=min z∈int(K)|K||Kz|, in the Banach-Mazur space of n-dimensional (classes of) convex bodies. Linear local stability in the neighborhood of the simplex is proved as well. The proof consists of an extension to the non-symmetric setting of methods that were recently introduced by Nazarov, Petrov, Ryabogin and Zvavitch, as well as proving results of independent interest concerning stability of square order of volumes of polars of non-symmetric convex bodies.",

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Local minimality of the volume-product at the simplex

Abstract

Bibliographical note

ASJC Scopus subject areas

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