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 is a set of polyhedra that satisfies the following conditions: 1. Every face of a polyhedron from is also in . 2. The intersection of any two polyhedra is a face of both and . Note that the empty set is a face of every polyhedron, and so the intersection of two polyhedra in 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 (polyhedral) 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 fan associated to a toric variety (see ). 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.
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