On Dvoretzky's theorem for subspaces of Lp

We prove that for any 2εE‖Z‖)≤Cexp⁡(−cmin⁡{αpε2n,(εn)2/p}),0<ε<1, where Z is the standard n-dimensional Gaussian vector, αp>0 is a constant depending only on p and C,c>0 are absolute constants. As a consequence we show optimal lower bound on the dimension of random almost spherical sections for these spaces. In particular, for any 20 is a constant depending only on p. This improves upon the previously known estimate due to Figiel, Lindenstrauss and Milman.