Revista Matemática Iberoamericana
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Published online: 2018-05-28
The variance conjecture on hyperplane projections of the $\ell_p^n$ balls
David Alonso[1] and Jesús Bastero[2] (1) Universidad de Zaragoza, Spain(2) Universidad de Zaragoza, Spain
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 $$\mathrm {Var} |X|^2 \leq C \mathrm {max}_{\xi\in S^{n-1}}\mathbb E\langle X,\xi\rangle^2\, \mathbb E|X|^2,$$ where $C$ 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.
Keywords: Variance conjecture, hyperplane projections, log-concave random vectors, convex bodies
Alonso David, Bastero Jesús: The variance conjecture on hyperplane projections of the $\ell_p^n$ balls. Rev. Mat. Iberoam. 34 (2018), 879-904. doi: 10.4171/RMI/1007