On Dvoretzky's theorem for subspaces of $L_p$

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: $$\mathbb P\left( \big| \|Z\| - \mathbb E\|Z\| \big| > \varepsilon \mathbb E\|Z\| \right) \leq C \exp \left (- c \min \left\{ \alpha_p \varepsilon^2 n, (\varepsilon n)^{2/p} \right\} \right), \quad 0<\varepsilon<1 ,$$ 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 $(1+\varepsilon)$-embedded into $X$ with $k\geq c_p \min\{ \varepsilon^2 n , (\varepsilon n)^{2/p}\}$, 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.