Polyhedral complex

A finite set of closed convex polytopes in a certain $\mathbf R^n$ 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 $|P|$ of a polyhedral complex $P$ is the union of all polytopes entering into it and is itself an (abstract) polyhedron (cf. Polyhedron, abstract). The number of polytopes in $P$ as a rule is less than the number of simplices in a triangulation. A polyhedral complex $P_1$ is called a subdivision of a complex $P$ if their underlying spaces coincide and if each polytope from $P_1$ lies in a certain polytope from $P$. A star-like subdivision of a complex $P$ with centre at a point $a\in|P|$ is obtained by means of a decomposition of the closed polytopes containing $a$ into cones with vertices at $a$ over those faces that do not contain $a$. Any polyhedral complex $P$ has a subdivision $K$ 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 $P$ with centres at all the vertices of $P$.

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For extra references see also Polyhedral chain and Simplicial complex.