Equidecomposable and weakly neighborly polytopes

A polytope is equidecomposable if all its triangulations have the same face numbers. For an equidecomposable polytope all minimal affine dependencies have an equal number of positive and negative coefficients. A subclass consists of the weakly neighborly polytopes, those for which every set of vertices is contained in a face of at most twice the dimension as the set. Theh-vector of every triangulation of a weakly neighborly polytope equals theh-vector of the polytope itself. Combinatorial properties of this class of polytopes are studied. Gale diagrams of weakly neighborly polytopes with few vertices are characterized in the spirit of the known Gale diagram characterization of Lawrence polytopes, a special class of weakly neighborly polytopes.