Small ball probability estimates in terms of width

Volume 169 / 2005

Rafał Latała, Krzysztof Oleszkiewicz 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.

Authors

  • Rafał LatałaInstitute of Mathematics
    Warsaw University
    Banacha 2
    02-097 Warszawa, Poland
    e-mail
  • Krzysztof OleszkiewiczInstitute of Mathematics
    Warsaw University
    Banacha 2
    02-097 Warsaw, Poland
    e-mail

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