Two theorems on Euclidean distance matrices and Gale transform

We present a characterization of those Euclidean distance matrices (EDMs) D which can be expressed as D = lambda (E - C) for some nonnegative scalar lambda and some correlation matrix C, where E is the matrix of all ones. This shows that the cones cone (E - E-n) not subset of or equal to cone (E - E-n) = D-n, where E-n is the elliptope (set of correlation matrices) and on is the (closed convex) cone of EDMs. The characterization is given using the Gale transform of the points generating D. We also show that given points p(1), p(2),...,p(n) is an element of R-r, for any scalars; lambda (1), lambda (2),...lambda (n),; such that we have n Sigma (j=1) lambda (j)p(j) = 0, Sigma (j=1) lambda (j) = 0, we have Sigma (n)(j=1) lambda (j)textbackslash{} textbackslash{}p(i) - p(j)textbackslash{} textbackslash{} (2) = alpha for all i = 1,..., n, for some scalar alpha independent of i. (C) 2002 Elsevier Science Inc. All rights reserved.