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arXiv · 1703.09973

The variance conjecture on projections of the cube

David Alonso-Gutiérrez·Julio Bernués
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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 $C$ $$\textrm{Var}_μ|x|^2\leq C \sup_{θ\in S^{n-1}}{\mathbb E}_μ\langle x,θ\rangle^2{\mathbb E}_μ|x|^2, $$ provided that $1\leq k\leq\sqrt n$. We also prove that if $1\leq k\leq n^{\frac{2}{3}}(\log n)^{-\frac{1}{3}}$, the conjecture is true for the family of uniform probabilities on its projections on random $(n-k)$-dimensional subspaces.

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David Alonso-Gutiérrez, Julio Bernués. 2017-03-29. The variance conjecture on projections of the cube. https://arxiv.org/abs/1703.09973

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