DISTANCE MATRICES AND REGULAR FIGURES

A regular figure (which includes all regular polygons) is a set of poi nts on a hypersphere whose center coincides with their centroid. We ch aracterize all regular figures as those whose points generate a Euclid ean distance matrix (EDM) with eigenvector e, the vector of all ones. Restricting the classical maps of Schoenberg, Gower, and Critchley for all EDMs to the subcone of EDMs with eigenvector e yields new geometr ical information about the generating points and a simple formula for the radius of the hypersphere.