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

TL;DR

This paper improves the known constants in anisotropic isoperimetric and Brunn-Minkowski inequalities for convex sets, reducing the dependence on dimension and proposing a conjecture for the optimal constant.

Contribution

It provides improved dimension-dependent constants in key geometric inequalities for convex sets and introduces new conjectures on their optimal bounds.

Findings

Improved constant from Cn^7 to Cn^6 for convex sets

Reduced constant from Cn^7 to Cn^5 for centrally symmetric convex sets

Conjecture that the optimal constant is proportional to n^2

Abstract

In this paper we revisit the anisotropic isoperimetric and the Brunn-Minkowski inequalities for convex sets. The best known constant $C(n)=Cn^{7}$ depending on the space dimension $n$ in both inequalities is due to Segal [\ref{bib:Seg.}]. We improve that constant to $Cn^6$ for convex sets and to $Cn^5$ for centrally symmetric convex sets. We also conjecture, that the best constant in both inequalities must be of the form $Cn^2,$ 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.