JournalsrmiVol. 34, No. 2pp. 879–904

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

  • David Alonso

    Universidad de Zaragoza, Spain
  • Jesús Bastero

    Universidad de Zaragoza, Spain
The variance conjecture on hyperplane projections of the $\ell_p^n$ balls cover

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Abstract

We show that for any 1p1\leq p\leq\infty, the family of random vectors uniformly distributed on hyperplane projections of the unit ball of pn\ell_p^n verify the variance conjecture

VarX2CmaxξSn1EX,ξ2EX2,\mathrm {Var} |X|^2 \leq C \mathrm {max}_{\xi\in S^{n-1}}\mathbb E\langle X,\xi\rangle^2\, \mathbb E|X|^2,

where CC depends on pp but not on the dimension nn 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 pn\ell_p^n-ball verify the variance conjecture.

Cite this article

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

DOI 10.4171/RMI/1007