Superconcentration, and randomized Dvoretzky's theorem for spaces with 1-unconditional bases

Abstract: Let $n$ be a sufficiently large natural number and let $B$ be an origin-symmetric convex body in $R^n$ in the $\ell$-position, and such that the normed space $(R^n,|\cdot|_B)$ admits a $1$-unconditional basis. Then for any $\varepsilon\in(0,1/2]$, and for random $c\varepsilon\log n/\log\frac{1}{\varepsilon}$-dimensional subspace $E$ distributed according to the rotation-invariant (Haar) measure, the section $B\cap E$ is $(1+\varepsilon)$-Euclidean with probability close to one. This shows that the "worst-case" dependence on $\varepsilon$ in the randomized Dvoretzky theorem in the $\ell$-position is significantly better than in John's position. It is a previously unexplored feature, which has strong connections with the concept of superconcentration introduced by S. Chatterjee. In fact, our main result follows from the next theorem: Let $B$ be as before and assume additionally that $B$ has a smooth boundary and ${\mathbb E}{\gamma_n}|\cdot|_B\leq n^c\,{\mathbb E}{\gamma_n}\big|{\rm grad}_B(\cdot)\big|_2$ for a small universal constant $c>0$, where ${\rm grad}_B(\cdot)$ is the gradient of $|\cdot|_B$ and $\gamma_n$ is the standard Gaussian measure in $R^n$. Then for any $p\in[1,c\log n]$ the $p$-th power of the norm $|\cdot|_B^p$ is $\frac{C}{\log n}$--superconcentrated in the Gauss space.