Small ball probability estimates in terms of width

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

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