Small ball probability estimates in terms of width
Volume 169 / 2005
Studia Mathematica 169 (2005), 305-314
MSC: Primary 60G15; Secondary 60E15.
DOI: 10.4064/sm169-3-6
Abstract
A certain inequality conjectured by Vershynin is studied. It is proved that for any symmetric convex body $K \subseteq {\mathbb R}^{n}$ with inradius $w$ and $\gamma_{n}(K) \leq 1/2$ we have $\gamma_{n}(sK) \leq (2s)^{w^{2}/4}\gamma_{n}(K)$ for any $s \in [0,1]$, where $\gamma_n$ is the standard Gaussian probability measure. Some natural corollaries are deduced. Another conjecture of Vershynin is proved to be false.
