AI 中文总结
本研究探讨高斯测度下广义体积乘积的极大化问题，证明极大化者为凸体，并给出低维精确形式及高维微分方程描述，同时明确欧几里得单位球唯一极大化的参数范围。
AI 中文摘要
我们研究了广义体积乘积\\[ \gamma_\sigma^n(A)\\,\gamma_\sigma^n(A^\circ) \\]在所有可测子集$A\subset\mathbb{R}^n$中的极大化问题，其中$A^\circ$表示$A$的极集，$\gamma_\sigma^n$表示$\mathbb{R}^n$上协方差为$\sigma^2 I_n$（$\sigma>0$）的中心高斯概率测度。结果表明，极大化者依赖于$\sigma$。我们证明了极大化者存在且为凸体。在维数$n=1$时，我们找到了极大化者的精确形式。在维数$n\ge 2$时，我们证明它们是光滑旋转体，其支撑函数满足某个微分方程。此外，对于$\sigma^2 \le \frac{1}{n}$，我们证明欧几里得单位球是唯一极大化者，而对于$\sigma^2\ge {\frac{2}{n+1}}$，情况不再如此。在维数$n=2$时，我们通过证明对于$\sigma^2 \le \frac{2}{3}$，欧几里得单位球是唯一极大化者来填补这一空白。
英文摘要
We study the maximizers of the generalized volume product \[ γ_σ^n(A)\,γ_σ^n(A^\circ) \] among all measurable subsets $A\subset\mathbb{R}^n$, where $A^\circ$ denotes the polar set of $A$, and where $γ_σ^n$ denotes the centered Gaussian probability measure on $\mathbb{R}^n$ with covariance $σ^2 I_n$, $σ>0$. It turns out that the maximizers depend on $σ$. We prove that they exist and are convex bodies. In dimension $n=1$, we find the exact form of the maximizers. In dimension $n\ge 2$, we show that they are smooth bodies of revolution whose support function satisfies a certain differential equation. Moreover, for $σ^2 \le \frac{1}{n}$ we show that the Euclidean unit ball is the unique maximizer, while this is no longer the case for $σ^2\ge {\frac{2}{n+1}}$. In dimension $n=2$, we close the gap by showing that the Euclidean unit ball is the unique maximizer for $σ^2 \le \frac{2}{3}$.
Comments19 pages
