高斯测度的Brunn–Minkowski不等式
The Brunn--Minkowski inequality for the Gaussian measure
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中文总结 AI 辅助
本文研究高斯测度下的Brunn–Minkowski不等式，通过Neumann问题解的Hessian能量逐射线局部化等方法，确定了不等式成立的最优指数$\alpha_\gamma(n)$的精确表达式。
中文摘要 AI 辅助
设$\gamma_n$为$\mathbb{R}^n$（$n\ge2$）上的标准高斯测度，$\alpha_\gamma(n)$是使得对所有包含原点的凸体$K,L\subset\mathbb{R}^n$及所有$\lambda\in[0,1]$，都有\n \\[\n \gamma_n(\lambda K+(1-\lambda)L)^{\alpha_\gamma(n)} \ge \lambda\gamma_n(K)^{\alpha_\gamma(n)} +(1-\lambda)\gamma_n(L)^{\alpha_\gamma(n)}\n \\]\n成立的最大数。本文证明了\n \\[\n \alpha_\gamma(n) =1-\frac{2}{n-1} \frac{\Gamma(\frac n2)^2}{\Gamma(\frac{n-1}{2})^2}.\n \\]\n证明的核心是对Neumann问题（诺依曼问题）解的Hessian能量进行逐射线的径向-切向局部化，将Neumann问题的源选择简化为一维优化。线段长度的单调性和Laguerre谱分析确定了精确的一维值，而平面端点情况则单独处理。
英文摘要
Let $γ_n$ be the standard Gaussian measure on $\mathbb{R}^n$, $n\ge2$, and let $α_γ(n)$ be the largest number for which \[ γ_n(λK+(1-λ)L)^{α_γ(n)} \ge λγ_n(K)^{α_γ(n)} +(1-λ)γ_n(L)^{α_γ(n)} \] holds for all convex bodies $K,L\subset\mathbb{R}^n$ containing the origin and all $λ\in[0,1]$. In this paper, we prove that \[ α_γ(n) =1-\frac{2}{n-1} \frac{Γ(\frac n2)^2}{Γ(\frac{n-1}{2})^2}. \] The core of the proof is a raywise radial--tangential localization of the Hessian energy of a solution of a Neumann problem, which reduces source selection of the Neumann problem to a one-dimensional optimization. Monotonicity in the segment length and Laguerre spectral analysis determine the sharp one-dimensional value, whereas the planar endpoint is treated separately.
