发表机构
Institute of Mathematics, Hunan University（湖南大学数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为凸体中随机点对距离分布建立Brunn-Minkowski理论，证明泛函满足不等式并刻画Wulff微分测度，进而得到球的刚性与Minkowski方程的联系。
AI 中文摘要
设$K\subset\mathbb{R}^n$为凸体，$X$和$Y$为$K$中独立的均匀随机点，并设$R_K=(|K|/\omega_n)^{1/n}$。我们考虑分布函数\\[ D_K(\rho)=\mathbb P\{|X-Y|/R_K<\rho\}, \\]及其齐次对应形式\begin{equation*} \mathcal J_\rho(K)=\iint_{K\times K}\mathbf{1}_{\{|x-y|<\rho R_K\}}dxdy. \end{equation*}因此$D_K(\rho)$是$K$中距离小于$\rho R_K$的有序点对所占比例，等价地，它是$K$的归一化协变图的径向分布函数。该泛函满足Brunn-Minkowski不等式，等号仅在位似体时成立，其微分给出第一个Minkowski不等式。我们刻画了作为Wulff微分出现的Borel测度：它们恰好是质心为零、不集中于大子球面上的非零有限正测度，且相应的体在平移意义下唯一。我们进一步证明了亚临界范围内球的固定体积刚性和平稳刚性，以及所得方程的正则性。在饱和情形，该方程即为经典的Minkowski方程。
英文摘要
Let $K\subset\mathbb{R}^n$ be a convex body, let $X$ and $Y$ be independent uniform points of $K$, and set $R_K=(|K|/ω_n)^{1/n}$. We consider the distribution function \[ D_K(ρ)=\mathbb P\{|X-Y|/R_K<ρ\}, \] and its homogeneous counterpart \begin{equation*} \mathcal J_ρ(K)=\iint_{K\times K}\mathbf{1}_{\{|x-y|<ρR_K\}}dxdy. \end{equation*} Thus $D_K(ρ)$ is the proportion of ordered pairs of points of $K$ whose distance is less than $ρR_K$, or equivalently the radial distribution function of the normalized covariogram of $K$. The functional satisfies a Brunn--Minkowski inequality, with equality precisely for homothetic bodies, and its differential gives a first Minkowski inequality. We characterize the Borel measures that occur as Wulff differentials: they are exactly the nonzero finite positive measures with zero centroid which are not concentrated on a great subsphere, and the corresponding body is unique up to translation. We further prove fixed-volume and stationary rigidity of the ball in the subcritical range, together with regularity for the resulting equation. At saturation, the equation is the classical Minkowski equation.
