论文标题
关于将切片问题减少到集中对称凸体的注释
A note on the reduction of the slicing problem to centrally symmetric convex bodies
论文作者
论文摘要
在本文中，我们获得了绝对常数$ c $的最佳价值，以使每个各向同性凸体$ k \ subseteq \ subseteq \ mathbb {r}^n $以下不平等现象（克拉塔格证明了这一点，并减少了超成本的猜想，以使中心对称的convex bodies满足： cl_ {k_ {n+2}（g_k）}。 $$这里$ l_k $表示$ k $，$ g_k $其covariogram函数的各向同性常数，它是log-concave，对于任何log-concove函数$ g $，$ k_ {n+2}（g）$都是与log-conconconcave函数$ g $相关的convex of the unip parters in unip partric partric partric partric。为了获得这种不平等，每当$ g $满足比原木concavity更好的凹面时，就可以获得该家族中的凸体之间的鲜明包含结果，因为$ g_k $是$ g_k $，实际上$ \ frac {1} {1} {n} $ - 倒置。
In this paper, we obtain the best possible value of the absolute constant $C$ such that for every isotropic convex body $K \subseteq \mathbb{R}^n$ the following inequality (which was proved by Klartag and reduces the hyperplane conjecture to centrally symmetric convex bodies) is satisfied: $$ L_K\leq CL_{K_{n+2}(g_K)}. $$ Here $L_K$ denotes the isotropic constant of $K$, $g_K$ its covariogram function, which is log-concave, and, for any log-concave function $g$, $K_{n+2}(g)$ is a convex body associated to the log-concave function $g$, which belongs to a uniparametric family introduced by Ball. In order to obtain this inequality, sharp inclusion results between the convex bodies in this family are obtained whenever $g$ satisfies a better type of concavity than the log-concavity, as $g_K$ is, indeed $\frac{1}{n}$-concave.
