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.
Definition
A polyhedral complex K {\displaystyle {\mathcal {K}}} is a set of polyhedra that satisfies the following conditions:
1. Every face of a polyhedron from K {\displaystyle {\mathcal {K}}} is also in K {\displaystyle {\mathcal {K}}}.
2. The intersection of any two polyhedra σ 1 , σ 2 ∈ K {\displaystyle \sigma _{1},\sigma _{2}\in {\mathcal {K}}} is a face of both σ 1 {\displaystyle \sigma _{1}} and σ 2 {\displaystyle \sigma _{2}}.
Note that the empty set is a face of every polyhedron, and so the intersection of two polyhedra in K {\displaystyle {\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 Toric variety § Fundamental theorem for toric geometry).
- 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.