Small ball probability estimates in terms of width

Abstract

A certain inequality conjectured by Vershynin is studied. It is proved that for any $n$-dimensional symmetric convex body $K$ with inradius $w$ and ${\gamma }_n(K)\le 1/2$ there is ${\gamma }_n(sK)\le (2s{)}^{w^2/4}{\gamma }_n(K)$ for any $s\in [0,1]$. Some natural corollaries are deduced. Another conjecture of Vershynin is proved to be false.

10 pages

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