The variance conjecture on hyperplane projections of l_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 $$\text{Var}|X{|}^2\le C\underset{\xi \in S^{n-1}}{\max }\mathbb{E}⟨X,\xi {⟩}^2\mathbb{E}|X{|}^2,$$ 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.