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

TL;DR

This work studies the deficit $\Delta(K,f)$ of the local log-Brunn-Minkowski inequality and proves a monotonicity principle for segment addition along $K_t=K+tI$ with $I$ an interval. The derivative decomposition yields a practical pathway to deduce LLBM for wider classes of bodies, culminating in a self-contained proof of LLBM for all zonoids via a double induction on dimension and the number of interval summands. Consequently, LLBM is preserved under Minkowski summation with any zonoid, and the approach illuminates equality cases for smooth bodies with $C^2$ support functions, showing equality only for dilates. Collectively, the results connect deficit monotonicity with spectral-geometry perspectives of LLBM and provide new geometric tools for the log-Brunn-Minkowski theory.

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.

Theorems & Definitions (23)

• Lemma 1.1
• proof
• Theorem 1.2
• Theorem 1.3
• Lemma 3.1
• proof
• proof : Proof of Theorem \ref{['Thm3']}
• Lemma 3.2
• proof
• Lemma 3.3