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Small ball probability estimates in terms of width
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