Random version of Dvoretzky's theorem in $\ell_p^n$

Grigoris Paouris, Petros Valettas, Joel Zinn

Published 2015-10-25Version 1

We study the dependence on $\varepsilon$ in the critical dimension $k(n, p, \varepsilon)$ that one can find random sections of the $\ell_p^n$-ball which are $(1+\varepsilon)$-spherical. For any fixed $n$ we give lower estimates for $k(n, p, \varepsilon)$ for all eligible values $p$ and $\varepsilon$, which agree with the sharp estimates for the extreme values $p = 1$ and $p = \infty$. In order to do so, we provide bounds for the gaussian concentration of the $\ell_p$-norm.