Cones and convex bodies with modular face lattices

Abstract

Dedicated to Claus M. Ringel on the occasion of his 60th birthday.Abstract. If a convex body C has modular and irreducible face lattice (and is not strictly convex), there is a face-preserving homeomorphism from C to a section of a cone of hermitian matrices over R, C, or H, or C has dimension 8, 14 or 26.1991 Mathematics Subject Classification. 52A20, 06C05, 51A05, 15A48.Recall that a projective space consists of a set P (the points) and a set L (the lines) so that (a) Each pair of points is contained in a unique line, (b) If a, b, c, d are distinct points and the lines ab and cd intersect, then the lines ac and bd intersect (c) Each line contains at least 3 points and there are at least 2 lines (d) Every chain of subspaces (also called flats) has finite length. The maximum length of a chain starting with a point is the projective dimension of the space.The flats of a projective space form an algebraic, atomic, irreducible, modular lattice. Conversely, any lattice with these properties is the lattice of flats of a