The variance conjecture on hyperplane projections of l_p^n balls

TL;DR

This paper proves the variance conjecture for random vectors on hyperplane projections of all balls in all spaces, extending to Steiner symmetrizations, with bounds independent of dimension.

Contribution

It establishes the variance conjecture for hyperplane projections of all balls and relates it to Steiner symmetrizations, broadening the class of convex bodies satisfying the conjecture.

Findings

Variance conjecture holds for hyperplane projections of all balls.

Variance conjecture extends to Steiner symmetrizations of all balls.

Bounds are independent of the ambient dimension.

Abstract

We show that for any $1\leq p\leq\infty$, the family of random vectors uniformly distributed on hyperplane projections of the unit ball of $\ell_p^n$ verify the variance conjecture $$ \textrm{Var}\,|X|^2\leq C\max_{\xi\in S^{n-1}}\mathbb{E}\langle X,\xi\rangle^2\mathbb{E}|X|^2, $$ 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.