Selected Open Problems in Discrete Geometry and Optimization

A list of questions and problems posed and discussed in September 2011 at the following consecutive events held at the Fields Institute, Toronto: Workshop on Discrete Geometry, Conference on Discrete Geometry and Optimization, and Workshop on Optimization. We hope these questions and problems will contribute to further stimulate the interaction between geometers and optimizers

On the nullspace, the rangespace and the characteristic polynomial of Euclidean distance matrices

AbstractWe present a characterization of the nullspace and the rangespace of a Euclidean distance matrix (EDM) D in terms of the vector of all ones, and in terms of the Gale subspace G(D) and the realization matrix P corresponding to D. This characterization is then used to compute the characteristic polynomial of D. We also present some results concerning EDMs generated by regular figures and EDMs generated by centrally symmetric points