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 CVarμ|x|2≤Csupθ∈Sn−1Eμ〈x,θ〉2Eμ|x|2, provided that 1≤k≤n. We also prove that if 1≤k≤n23(logn)−13, the conjecture is true for the family of uniform probabilities on its projections on random (n−k)-dimensional subspaces.
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