Polyhedral complex
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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.
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 (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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