  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.
