Monotonicity of the deficit in the local log-Brunn-Minkowski inequality

TL;DR

This paper proves a monotonicity property of the deficit in the local log-Brunn-Minkowski inequality, showing how it behaves under addition of line segments and deriving implications for zonoids and equality cases.

Contribution

It establishes a monotonicity property of the deficit in the local log-Brunn-Minkowski inequality and derives new results for zonoids and equality conditions.

Findings

Monotonicity of the deficit under addition of line segments.

Extension of the inequality to zonoids.

Characterization of equality cases for smooth convex bodies.

Abstract

We establish a monotonicity property of the deficit associated with the local log-Brunn-Minkowski inequality (LLBM) under addition of line segments. As a corollary, if the LLBM holds for a convex body K, then it also holds for K+Z for any zonoid Z, which in particular yields a new proof of the inequality for zonoids. Moreover, assuming the LLBM is valid, we prove that equality in the LLBM for smooth convex bodies with $C^2$ support functions occurs only for homothetic bodies.