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

TL;DR

This paper investigates the critical dimension for random sections of the _p^n-ball to be nearly spherical, providing estimates that match known results at the extremes and analyzing _p-norm concentration.

Contribution

It offers new bounds on the critical dimension for _p^n-balls, extending understanding across all p and _epsilon, with tight concentration bounds for _p-norms.

Findings

Derived bounds for critical dimension k(n,p,_epsilon) for all p and _epsilon.

Matched estimates with known results at p=1 and p=_infinity.

Provided tight Gaussian concentration bounds for _p-norms.

Abstract

We study the dependence on $\varepsilon$ in the critical dimension $k(n,p,\varepsilon)$ for which one can find random sections of the $\ell_p^n$-ball which are $(1+\varepsilon)$-spherical. We give lower (and upper) estimates for $k(n,p,\varepsilon)$ for all eligible values $p$ and $\varepsilon$ as $n\to \infty$, which agree with the sharp estimates for the extreme values $p=1$ and $p=\infty$. Toward this end, we provide tight bounds for the Gaussian concentration of the $\ell_p$-norm.