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 $\gamma_{n}(K) \leq 1/2$ there is $\gamma_{n}(sK) \leq (2s)^{w^{2}/4}\gamma_{n}(K)$ for any $s \in [0,1]$. Some natural corollaries are deduced. Another conjecture of Vershynin is proved to be false. 
Rafa{\l} Lata{\l}a, Krzysztof Oleszkiewicz