Iso edge domains

Abstract Iso-edge domains are a variant of the iso-Delaunay decomposition introduced by Voronoi. They were introduced by Baranovskii & Ryshkov in order to solve the covering problem in dimension 5. In this work we revisit this decomposition and prove the following new results: • [(1)] We review the existing theory and give a general mass-formula for the iso-edge domains. • [(2)] We prove that the associated toroidal compactification of the moduli space of principally polarized abelian varieties is projective. • [(3)] We prove the Conway–Sloane conjecture in dimension 5. • [(4)] We prove that the quadratic forms for which the conorms are non-negative are exactly the matroidal ones in dimension 5.