The variance conjecture on hyperplane projections of the ${\ell }_p^n$ balls

Abstract

We show that for any $1\le p\le \infty$, the family of random vectors uniformly distributed on hyperplane projections of the unit ball of ${\ell }_p^n$ verify the variance conjecture

where $C$ depends on $p$ but not on the dimension $n$ or the hyperplane. We will also show a general result relating the variance conjecture for a random vector uniformly distributed on an isotropic convex body and the variance conjecture for a random vector uniformly distributed on any Steiner symmetrization of it. As a consequence we will have that the class of random vectors uniformly distributed on any Steiner symmetrization of an ${\ell }_p^n$-ball verify the variance conjecture.