Quantitative anisotropic isoperimetric and Brunn−Minkowski inequalities for convex sets with improved defect estimates

EPFL, Ecole Polytechnique Federale de Lausanne, Lausanne, CH-1015 Lausanne, Suisse

^a Corresponding author: davit.harutyunyan@epfl.ch

Received: 29 April 2016
Revised: 10 November 2016
Accepted: 17 January 2017

Abstract

In this paper we revisit the anisotropic isoperimetric and the Brunn−Minkowski inequalities for convex sets. The best known constant C(n) = Cn⁷ depending on the space dimension n in both inequalities is due to Segal [A. Segal, Lect. Notes Math., Springer, Heidelberg 2050 (2012) 381–391]. We improve that constant to Cn⁶ for convex sets and to Cn⁵ for centrally symmetric convex sets. We also conjecture, that the best constant in both inequalities must be of the form Cn², i.e., quadratic in n. The tools are the Brenier’s mapping from the theory of mass transportation combined with new sharp geometric-arithmetic mean and some algebraic inequalities plus a trace estimate by Figalli, Maggi and Pratelli.