关于对数凹旋转不变测度的多面体的最大表面积
Maximal surface area of polytopes with respect to log-concave rotation invariant measures
摘要：
在\cite{GL}中证明了$\mathbb{R}^n$中关于旋转不变对数凹概率测度$γ$的凸集的最大表面积是$\frac{\sqrt{n}}{\sqrt[4]{Var|X|} \sqrt{\mathbb{E}|X|}}$阶，其中$X$是$\mathbb{R}^n$中关于$γ$分布的随机向量。本文讨论了凸多面体$K$的表面积。我们发现$K$的$P_K$的最大表面积上紧的界限。我们表明，$γ(\partial P_K)\lesssim \frac{\sqrt{n}}{\mathbb{E}|X|}\cdot\sqrt{\log K}\cdot\log n$的所有$K$。这个界比所有$K\in [2,e^{\frac{c}{\sqrt{Var|X|}}}]$的一般界更好。此外，对于在该范围内的所有$K$，边界精确到$\log n$的因子：对于每个$K\in [2,e^{\frac{c}{\sqrt{Var|X|}}}]$，存在至多具有$K$小面的多面体$P_K$，使得$γ(\partial P_K)\gtrsim \frac{\sqrt{n}}{\mathbb{E}|X|}\sqrt{\log K}.$ %对于具有密度$C_{n,p} e^{-\frac{|y|^p}{p}}$（其中$p>0$）的测度$γ_p$，我们获得：$γ_p(\partial P_K)\lesssim \frac{\sqrt{n}}{\mathbb{E}X}\sqrt{\log K},$，其由F对标准高斯测度$γ_2$获得。纳扎罗夫
英文摘要：
It was shown in \cite{GL} that the maximal surface area of a convex set in $\mathbb{R}^n$ with respect to a rotation invariant log-concave probability measure $γ$ is of order $\frac{\sqrt{n}}{\sqrt[4]{Var|X|} \sqrt{\mathbb{E}|X|}}$, where $X$ is a random vector in $\mathbb{R}^n$ distributed with respect to $γ$. In the present paper we discuss surface area of convex polytopes $P_K$ with $K$ facets. We find tight bounds on the maximal surface area of $P_K$ in terms of $K$. We show that $γ(\partial P_K)\lesssim \frac{\sqrt{n}}{\mathbb{E}|X|}\cdot\sqrt{\log K}\cdot\log n$ for all $K$. This bound is better then the general bound for all $K\in [2,e^{\frac{c}{\sqrt{Var|X|}}}]$. Moreover, for all $K$ in that range the bound is exact up to a factor of $\log n$: for each $K\in [2,e^{\frac{c}{\sqrt{Var|X|}}}]$ there exists a polytope $P_K$ with at most $K$ facets such that $γ(\partial P_K)\gtrsim \frac{\sqrt{n}}{\mathbb{E}|X|}\sqrt{\log K}.$ %For the measures $γ_p$ with densities $C_{n,p} e^{-\frac{|y|^p}{p}}$ (where $p>0$) we obtain: $γ_p(\partial P_K)\lesssim \frac{\sqrt{n}}{\mathbb{E}X}\sqrt{\log K},$ which was obtained for the standard Gaussian measure $γ_2$ by F. Nazarov.
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