关于q>n时q阶对偶曲率积分的Brunn-Minkowski不等式
On the Brunn-Minkowski inequality for $q$-th dual quermassintegrals with $q>n$
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中文总结 AI 辅助
本文针对q>n时q阶对偶曲率积分的Brunn-Minkowski不等式展开研究，证明其对任意凸体及原点对称凸体的失效情形，建立原点对称凸体端点情形与无条件凸体全范围的不等式，并推导对偶曲率测度的唯一性结果。
中文摘要 AI 辅助
本文研究q>n时q阶对偶曲率积分的Brunn-Minkowski不等式，该问题由Sadovsky与Zhang新近提出。首先，通过二阶变分论证与降维构造，证明当q>n时该不等式对任意凸体均不成立，当q>n+2时甚至在原点对称凸体类中也不成立。其次，利用极惯性矩的Hadwiger不等式，证明原点对称凸体的端点情形q=n+2成立。最后，对无条件凸体，通过奇异加权Reilly公式与坐标切片Hardy不等式，在0<q≤n+1的全范围内建立该不等式。作为应用，推导了对应对偶曲率测度的若干唯一性结果。
英文摘要
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.
