Polyhedral complex

A finite set of closed convex polytopes in a certain which together with each polytope contains all its faces and is such that the intersection between the polytopes is either empty or is a face of each of them. An example of a polyhedral complex is the set of all vertices, edges and two-dimensional faces of the standard three-dimensional cube. One considers also complexes consisting of an infinite but locally finite family of polytopes. The concept of a polyhedral complex generalizes the concept of a geometric simplicial complex. The underlying space of a polyhedral complex is the union of all polytopes entering into it and is itself an (abstract) polyhedron (cf. Polyhedron, abstract). The number of polytopes in as a rule is less than the number of simplices in a triangulation. A polyhedral complex is called a subdivision of a complex if their underlying spaces coincide and if each polytope from lies in a certain polytope from . A star-like subdivision of a complex with centre at a point is obtained by means of a decomposition of the closed polytopes containing into cones with vertices at over those faces that do not contain . Any polyhedral complex has a subdivision that is a geometric simplicial complex. Such a subdivision can be obtained without adding new vertices. It is sufficient, for example, to carry out in sequence the star-like subdivisions of with centres at all the vertices of .

References

[1]	P.S. Aleksandrov, "Combinatorial topology" , Graylock , Rochester (1956) (Translated from Russian)

Comments

For extra references see also Polyhedral chain and Simplicial complex.