矩比较、Sudakov不等式与质心体的熵
Moment comparisons, Sudakov inequalities and entropy of centroid bodies
浏览论文内容
中文总结 AI 辅助
本文针对迷向对数凹随机向量与标准高斯向量，推导任意矩阶的规范函数矩比较估计，得到$L_p$-Sudakov相关结果与质心体的熵、范数等多类几何估计。
中文摘要 AI 辅助
设$X$是$\mathbb{R}^n$迷向对数凹随机向量，$G$是标准高斯向量。从规范函数的一阶矩比较出发，我们推导了任意矩阶的对应估计。对任意规范函数$\phi$和任意$q\geqslant 1$，有$$ \\|\phi(G)\\|_q\leqslant C\sqrt{\ln(en)+q}\\,\\|\phi(X)\\|_q,\qquad \\|\phi(X)\\|_q\leqslant C\left(\sqrt{\ln(en)}+\psi(X)\sqrt q\right)\\|\phi(G)\\|_q. $$将其应用于支撑函数，得到$L_2$-Sudakov常数的精确最坏情况阶$C\sqrt{\ln(en)}$以及定量$L_p$-Sudakov估计。在Latala二进链的每一层，结合本文所用三种定量$L_p$-Sudakov估计中的最强形式，可得：当$Y$的弱矩被$X$的弱矩控制时，$$ \left(\mathbb{E}\\|Y\\|^p\right)^{1/p}\leqslant C\left(n^{1/4}\sqrt{\ln(en)}\\,\ln(e+\ln(en))\\,\mathbb{E}\\|X\\|+\sigma_p(Y)\right) $$。另一方面，我们研究了与$Z_r(X)^\circ$相关的自生成度量的广义对偶Sudakov问题，证明了$Z_p(X)$的无维数填充估计。我们还得到了质心体的平均范数估计、通过任意对称凸体的分解以及仿射维数细化结果。
英文摘要
Let $X$ be an isotropic log-concave random vector in $\mathbb{R}^n$ and let $G$ be standard Gaussian. Starting from the first moment comparison for gauges, we derive corresponding estimates at arbitrary moment orders. For every gauge $ϕ$ and every $q\geqslant 1$, $$ \|ϕ(G)\|_q\leqslant C\sqrt{\ln(en)+q}\,\|ϕ(X)\|_q,\qquad \|ϕ(X)\|_q\leqslant C\left(\sqrt{\ln(en)}+ψ(X)\sqrt q\right)\|ϕ(G)\|_q. $$ Applied to support functions, this gives the sharp worst case order $C\sqrt{\ln(en)}$ for the $L_2$-Sudakov constant and quantitative $L_p$-Sudakov estimates. Combining, at each level of Latala's dyadic chain, the strongest of the three quantitative $L_p$-Sudakov estimates used here yields $$ \left(\mathbb{E}\|Y\|^p\right)^{1/p}\leqslant C\left(n^{1/4}\sqrt{\ln(en)}\,\ln(e+\ln(en))\,\mathbb{E}\|X\|+σ_p(Y)\right) $$ whenever the weak moments of $Y$ are dominated by those of $X$. In a second direction, we study the generalized dual Sudakov problem for the self generated metrics associated with $Z_r(X)^\circ$ and prove dimension free packing estimates for $Z_p(X)$. We also obtain mean norm estimates for centroid bodies, factorization through arbitrary symmetric convex bodies and an affine dimensional refinement.
