Mean width inequalities of sections and projections for isotropic measures

TL;DR

This paper extends mean width inequalities for convex bodies under isotropic measures, providing new equality conditions and employing advanced geometric and measure transportation techniques.

Contribution

It introduces a novel proof approach for mean width inequalities using Ball-Barthe inequality and isotropic embedding, extending prior results.

Findings

Established mean width inequalities with complete equality conditions.

Extended recent work by Alonso-Gutiérrez and Brazitikos.

Utilized advanced tools like mass transportation and isotropic embedding.

Abstract

In this paper, we establish mean width inequalities of sections and projections of convex bodies for isotropic measures with complete equality conditions, which extends the recent work of Alonso-Guti\'{e}rrez and Brazitikos. Different from their approach, our proof is based on the approach developed by Lutwak, Yang and Zhang, by using the Ball-Barthe inequality, the mass transportation, and the isotropic embedding.