A Remark on the Rank of Positive Semidefinite Matrices Subject to Affine Constraints

LetKn be the cone of positive semidefinite n£n matrices and letA be an affine subspace of the space of symmetric matrices such that the intersectionKn \A is nonempty and bounded. Suppose that n ‚ 3 and that codimA D ¡rC22 ¢ for some 1 • r • n ¡ 2. Then there is a matrix X 2 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