A remark on the rank of positive semidefinite matrices subject to affine constraints

Abstract

Let Kn be the cone of positive semidefinite n × n matrices and let A be an affine subspace of the space of symmetric matrices such that the intersection Kn ∩ A is nonempty and bounded. Suppose that n ≥ 3 and that codim A = (r+22) for some 1 ≤ r ≤ n - 2. Then there is a matrix X ∈ Kn ∩ A such that rank X ≤ r. We give a short geometric proof of this result, use it to improve a bound on realizability of weighted graphs as graphs of distances between points in Euclidean space, and describe its relation to theorems of Bohnenblust, Friedland and Loewy, and Au-Yeung and Poon.