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LOCAL MINIMALITY OF THE VOLUME-PRODUCT AT THE SIMPLEX

Part of:General convexity

Published online by Cambridge University Press:  13 December 2010

Jaegil Kim and
Shlomo Reisner
Jaegil Kim
Affiliation:
Department of Mathematics, Kent State University, Kent, OH 44242, U.S.A. (email: jkim@math.kent.edu)
Shlomo Reisner
Affiliation:
Department of Mathematics, University of Haifa, Haifa 31905, Israel (email: reisner@math.haifa.ac.il)

Abstract

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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.

MSC classification

Secondary:52A40: Inequalities and extremum problems

Information

Type
Research Article
Information
Mathematika , Volume 57 , Issue 1 , January 2011, pp. 121 - 134
Copyright
Copyright © University College London 2011

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