# The variance conjecture on hyperplane projections of the $\ell_p^n$ balls

### David Alonso

Universidad de Zaragoza, Spain### Jesús Bastero

Universidad de Zaragoza, Spain

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## 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

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.

## Cite this article

David Alonso, Jesús Bastero, The variance conjecture on hyperplane projections of the $\ell_p^n$ balls. Rev. Mat. Iberoam. 34 (2018), no. 2, pp. 879–904

DOI 10.4171/RMI/1007