The variance conjecture on hyperplane projections of l_pⁿ balls

2016/10/13 by Alonso-Gutiérrez, David, Bastero, Jesús
#FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1610.04023

Abstract

We show that for any 1≤ p≤∞, the family of random vectors uniformly distributed on hyperplane projections of the unit ball of ℓ_pⁿ verify the variance conjecture \textrmVar |X|²≤ Cmax_{ξ∈ S^n-1}𝔼⟨ X,ξ⟩²𝔼|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 ℓ_pⁿ-ball verify the variance conjecture.

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