Superconcentration, and randomized Dvoretzky's theorem for spaces with 1unconditional bases
Abstract
Let $n$ be a sufficiently large natural number and let $B$ be an originsymmetric 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 rotationinvariant (Haar) measure, the section $B\cap E$ is $(1+\varepsilon)$Euclidean with probability close to one. This shows that the "worstcase" 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2017
 arXiv:
 arXiv:1702.00859
 Bibcode:
 2017arXiv170200859T
 Keywords:

 Mathematics  Metric Geometry;
 Mathematics  Probability
 EPrint:
 removed the smoothness assumption in the main theorem