Extremal Properties for Dissections of Convex 3-Polytopes

Abstract

A dissection of a convex d-polytope is a partition of the polytope into d-simplices whose vertices are among the vertices of the polytope. Triangulations are dissections that have the additional property that the set of all its simplices forms a simplicial complex. The size of a dissection is the number of d-simplices it contains. This paper compares triangulations of maximal size with dissections of maximal size. We also exhibit lower and upper bounds for the size of dissections of a 3-polytope and analyze extremal size triangulations for specific nonsimplicial polytopes: prisms, antiprisms, Archimedean solids, and combinatorial d-cubes.