On the $\psi_2$-behaviour of linear functionals on isotropic convex bodies

G. Paouris ¹

¹ Department of Mathematics University of Crete Iraklion 714-09, Crete, Greece

Studia Mathematica, Tome 168 (2005) no. 3, pp. 285-299

Résumé

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 $r\simeq\sqrt{n}\,L_K$.

Détail

DOI : 10.4064/sm168-3-7

Keywords: slicing problem reduced study isotropic convex bodies mathop diam nolimits leq sqrt where isotropic constant study psi behaviour linear functionals class bodies proved langle cdot theta rangle psi leq theta subset n measure sigma geq exp c sqrt however there exist isotropic convex bodies uniformly bounded geometric distance euclidean ball max theta n langle cdot theta rangle psi geq sqrt different direction average psi behaviour linear functionals isotropic convex body implies strong dimension dependent concentration volume inside ball radius simeq sqrt