A sharp combinatorial version of Vaaler's theorem

Abstract

In 1979 Vaaler proved that every d-dimensional central section of the cube [−1, 1]n has volume at least 2d. We prove the following sharp combinatorial analogue. Let K be a d-dimensional subspace of ℝn. Then, there exists a probability measure P on the section [−1, 1]n ∩ K such that the quadratic form dominates the identity on K (in the sense that the difference is positive semi-definite).