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

2025/10/21 by Shouda Wang, Wang, Shouda · 1 citation
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Metric Geometry (math.MG) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2510.19061

openalex publication_date 2025/10/21 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

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 C² support functions occurs only for homothetic bodies.

Citations

• On the uniqueness of even L^p Minkowski problem
• The Reverse-log-Brunn-Minkowski inequality
• The Logarithmic Minkowski conjecture and the L_p-Minkowski Problem
• The local logarithmic Brunn-Minkowski inequality for zonoids
• Log-Brunn-Minkowski inequality under symmetry
• Equivalence of the local and global versions of the L^p-Brunn-Minkowski inequality
• Local L^p-Brunn-Minkowski inequalities for p < 1
• A letter: The log-Brunn-Minkowski inequality for complex bodies
• Remarks on the conjectured log-Brunn-Minkowski inequality
• Remarks on the conjectured log-Brunn–Minkowski inequality

Cited by

• On the uniqueness of even L^p Minkowski problem

Related

• Remarks on the conjectured log-Brunn-Minkowski inequality
• A note on the quantitative local version of the log-Brunn-Minkowski inequality
• The local logarithmic Brunn-Minkowski inequality for zonoids
• Local L^p-Brunn-Minkowski inequalities for p < 1
• The Reverse-log-Brunn-Minkowski inequality