On Dvoretzky's theorem for subspaces of $L_p$

We prove that for any $2<p<\mathrm{\infty }$ 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: $$\mathbb{P}(\mid \mathrm{\parallel }Z\mathrm{\parallel }-\mathbb{E}\mathrm{\parallel }Z\mathrm{\parallel }\mid >\epsilon \mathbb{E}\mathrm{\parallel }Z\mathrm{\parallel })\le C\exp (-c\min \{{\alpha }_p{\epsilon }^{2}n,(\epsilon n{)}^{2\mathrm{/}p}\}),0<\epsilon <1,$$ 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 $2<p<\mathrm{\infty }$ and every $n$-dimensional subspace $X$ of $L_p$, the Euclidean space ${\mathrm{\ell }}_2^k$ can be $(1+\epsilon )$-embedded into $X$ with $k\geq c_p \min{ \varepsilon^2 n , (\varepsilon n)^{2/p}}$,where$c_p>0$isaconstantdependingonlyon$p$. This improves upon the previously known estimate due to Figiel, Lindenstrauss and V. Milman.