The local logarithmic Brunn-Minkowski inequality for bodies of revolution

TL;DR

This paper proves a local version of the logarithmic Brunn-Minkowski inequality specifically for bodies of revolution, extending the result to asymmetric cases and analyzing equality conditions using operator theory and spherical harmonics.

Contribution

It introduces a novel proof technique combining operator theory and spherical function decomposition for bodies of revolution, including asymmetric cases.

Findings

Proves the local logarithmic Brunn-Minkowski inequality for bodies of revolution.

Generalizes the inequality to asymmetric bodies of revolution.

Analyzes equality cases using spherical harmonic decomposition.

Abstract

We prove the local logarithmic Brunn-Minkowski inequality for bodies of revolution. Furthermore, we give a generalization for one origin symmetric body of revolution and one body of revolution that does not need to be symmetric and restrict possible equality cases. The proof uses an operator theoretic approach together with the decomposition of spherical functions into isotypical components with respect to rotations around a fixed axis.