A sharp combinatorial version of Vaaler's theorem

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