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 $C(n)=C{n}^7$ depending on the space dimension $n$ in both inequalities is due to Segal [\ref{bib:Seg.}]. We improve that constant to $C{n}^6$ for convex sets and to $C{n}^5$ for centrally symmetric convex sets. We also conjecture, that the best constant in both inequalities must be of the form $C{n}^2,$ 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.

arXiv commentRecently Emanuel Milman from Technion pointed out a mistake, where I missed an n^{1/2} on the right hand side of (3.18). The correction of the mistake leads to a reproduction of the already known (Legal [35]) constant Cn^7