The variance conjecture on projections of the cube

2017/03/29 by Alonso-Gutiérrez, David, Bernués, Julio
#FOS: Mathematics #Functional Analysis (math.FA) #Primary 52B09 #Secondary 52A23

paper · doi:10.48550/arxiv.1703.09973

Abstract

We prove that the uniform probability measure μ on every (n-k)-dimensional projection of the n-dimensional unit cube verifies the variance conjecture with an absolute constant C \textrmVar_μ|x|²≤ C sup_{θ∈ S^n-1}\mathbb E_μ⟨ x,θ⟩²\mathbb E_μ|x|², provided that 1≤ k≤√ n. We also prove that if 1≤ k≤ n^(2)/(3)(log n)^-(1)/(3), the conjecture is true for the family of uniform probabilities on its projections on random (n-k)-dimensional subspaces.

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