On the $\psi_2$-behaviour of linear functionals on isotropic convex bodies
Volume 168 / 2005
Abstract
The slicing problem can be reduced to the study of isotropic convex bodies $K$ with $\mathop{\rm diam}\nolimits (K)\leq c\sqrt{n}\,L_K$, where $L_K$ is the isotropic constant. We study the $\psi_2$-behaviour of linear functionals on this class of bodies. It is proved that $\|\langle \cdot ,\theta\rangle\|_{\psi_2}\leq CL_K$ for all $\theta $ in a subset $U$ of $S^{n-1}$ with measure $\sigma (U)\geq 1-\exp (-c\sqrt{n})$. However, there exist isotropic convex bodies $K$ with uniformly bounded geometric distance from the Euclidean ball, such that $\max_{\theta\in S^{n-1}}\|\langle \cdot ,\theta\rangle\|_{\psi_2} \geq c\sqrt[4]{n}\,L_K$. In a different direction, we show that good average $\psi_2$-behaviour of linear functionals on an isotropic convex body implies very strong dimension-dependent concentration of volume inside a ball of radius $r\simeq\sqrt{n}\,L_K$.
