Random version of Dvoretzky's theorem in ℓpn

We study the dependence on ε in the critical dimension k(n,p,ε) for which one can find random sections of the ℓpn-ball which are (1+ε)-spherical. We give lower (and upper) estimates for k(n,p,ε) for all eligible values p and ε as n→∞, which agree with the sharp estimates for the extreme values p=1 and p=∞. Toward this end, we provide tight bounds for the Gaussian concentration of the ℓp-norm.