On the Brunn-Minkowski inequality for $q$-th dual quermassintegrals with $q>n$

Haizhong Li; Yao Wan; Yijia Zhang

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arXiv · 2608.03949

On the Brunn-Minkowski inequality for $q$-th dual quermassintegrals with $q>n$

Haizhong Li·Yao Wan·Yijia Zhang

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Abstract

In this paper, we study the Brunn-Minkowski inequality for $q$-th dual quermassintegrals with $q>n$. This problem was recently posed by Sadovsky and Zhang. First, by a second variation argument and a dimension reduction construction, we show that the inequality fails for arbitrary convex bodies when $q>n$, and fails even in the origin-symmetric class when $q>n+2$. Secondly, we prove the endpoint case $q=n+2$ for origin-symmetric convex bodies via Hadwiger's inequality for the polar moment of inertia. Finally, for unconditional convex bodies, we establish the inequality in the full range $0<q\le n+1$ by using a singular weighted Reilly formula and a coordinate-slice Hardy inequality. As applications, we derive several uniqueness results for the corresponding dual curvature measures.

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BibTeX RIS

Haizhong Li, Yao Wan, Yijia Zhang. 2026-08-04. On the Brunn-Minkowski inequality for $q$-th dual quermassintegrals with $q>n$. https://arxiv.org/abs/2608.03949

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