On Dvoretzky's theorem for subspaces of $L_p$

Abstract

We prove that for any $2<p<\infty$ and for every $n$-dimensional subspace $X$ of $L_p$, represented on ${\mathbb{R}}^n$, whose unit ball $B_X$ is in Lewis' position one has the following two-level Gaussian concentration inequality: \[ \mathbb P\left( \big| \|Z\| - \mathbb E\|Z\| \big| > \varepsilon \mathbb E\|Z\| \right) \leq C \exp \left (- c \min \left\{ α_p \varepsilon^2 n, (\varepsilon n)^{2/p} \right\} \right), \quad 0<\varepsilon<1 , \] where $Z$ is a standard $n$-dimensional Gaussian vectors, ${\alpha }_p>0$ is a constant depending only on $p$ and $C,c>0$ are absolute constants. As a consequence we show optimal lower bound for the dimension of almost spherical sections for these spaces. In particular, for any $2<p<\infty$ and every $n$-dimensional subspace $X$ of $L_p$, the Euclidean space ${\ell }_2^k$ can be $(1+\epsilon )$-embedded into $X$ with $k\ge c_p\min \{{\epsilon }^{2}n,(\epsilon n{)}^{2/p}\}$, where $c_p>0$ is a constant depending only on $p$. This improves upon the previously known estimate due to Figiel, Lindenstrauss and V. Milman.

25 pages; minor changes

References in corpus (1)

• Random version of Dvoretzky's theorem in ${\ell }_p^n$