论文标题
通过各向同性的反向Loomis-Whitney不平等现象
Reverse Loomis-Whitney inequalities via isotropicity
论文作者
论文摘要
给定一个中心的凸面$ k \ subseteq \ mathbb {r}^n $，我们研究常数$ \tildeλ（k）$的最佳价值，使得存在正常的基础$ \ \ {w_i \ \} _ {i = 1}^n $：以下是以下的反向dual loomis-exequ | k |^{n-1} \ leqslant \tildeλ（k）\ prod_ {i = 1}^n | k \ cap w_i^\ perp |。 $$我们证明，对于某些绝对$ c> 1 $，$ \tildeλ（k）\ leqslant（cl_k）^n $，并且根据$ l_k $的估计，$ k $的$ k $的估计值是$ k $的$ k $，在某种意义上是一个绝对的常数$ c> 1 $ and convex $ c $ convex $ c $ $（cl_k）^n \ leqslant \tildeλ（k）\ leqslant（cl_k）^n $。我们还证明了更一般的反向双重织机 - 惠特尼的不平等以及loomis-whitney和双重Loomis-Whitney不平等的反向限制版本。
Given a centered convex body $K\subseteq\mathbb{R}^n$, we study the optimal value of the constant $\tildeΛ(K)$ such that there exists an orthonormal basis $\{w_i\}_{i=1}^n$ for which the following reverse dual Loomis-Whitney inequality holds: $$ |K|^{n-1}\leqslant \tildeΛ(K)\prod_{i=1}^n|K\cap w_i^\perp|. $$ We prove that $\tildeΛ(K)\leqslant(CL_K)^n$ for some absolute $C>1$ and that this estimate in terms of $L_K$, the isotropic constant of $K$, is asymptotically sharp in the sense that there exists another absolute constant $c>1$ and a convex body $K$ such that $(cL_K)^n\leqslant\tildeΛ(K)\leqslant(CL_K)^n$. We also prove more general reverse dual Loomis-Whitney inequalities as well as reverse restricted versions of Loomis-Whitney and dual Loomis-Whitney inequalities.
