Polyhedral complex

In mathematics, a polyhedral complex is a set of polyhedra in a real vector space that fit together in a specific way. Polyhedral complexes generalize simplicial complexes and arise in various areas of polyhedral geometry, such as tropical geometry, splines and hyperplane arrangements.

Definition
A polyhedral complex $$\mathcal{K}$$ is a set of polyhedra that satisfies the following conditions:
 * 1. Every face of a polyhedron from $$\mathcal{K}$$ is also in $$\mathcal{K}$$.
 * 2. The intersection of any two polyhedra $$\sigma_1, \sigma_2 \in \mathcal{K}$$ is a face of both $$\sigma_1$$ and $$\sigma_2$$.

Note that the empty set is a face of every polyhedron, and so the intersection of two polyhedra in $$\mathcal{K}$$ may be empty.

Examples

 * Tropical varieties are polyhedral complexes satisfying a certain balancing condition.
 * Simplicial complexes are polyhedral complexes in which every polyhedron is a simplex.
 * Voronoi diagrams.
 * Splines.

Fans
A fan is a polyhedral complex in which every polyhedron is a cone from the origin. Examples of fans include:
 * The normal fan of a polytope.
 * The Gröbner fan of an ideal of a polynomial ring.
 * A tropical variety obtained by tropicalizing an algebraic variety over a valued field with trivial valuation.
 * The recession fan of a tropical variety.