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

Shouda Wang

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arXiv · 2510.19061

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

Shouda Wang

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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.

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Shouda Wang. 2025-10-21. Monotonicity of the deficit in the local log-Brunn-Minkowski inequality. https://arxiv.org/abs/2510.19061

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