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

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.