Abstract:An abstract
convex geometry describes the combinatorics of the convex subsets of a
finite point set in Euclidean space. A complex of oriented matroids
describes the combinatorics of a polyhedron sliced by hyperplanes in
Euclidean space. It is untractable to decide whether a given abstract
convex geometry or a given complex of oriented matroid can be realized
by a point set or by a sliced polyhedron, respectively. We show that
every abstract convex geometry can be realized by a sliced polyhedron, i.e., abstract
convex geometries are realizable complexes of oriented matroids. We
further discuss extensions to larger classes and bounds on the dimension
of the realization.
Joint work with Victor Chepoi and Jérémie Chalopin
