Polyhedral complex
Math concept
In mathematics, a polyhedral complex is a set of polyhedra in a real vector space that fit together in a specific way.[1] 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.[2]
- 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.[3][4]
- A tropical variety obtained by tropicalizing an algebraic variety over a valued field with trivial valuation.
- The recession fan of a tropical variety.
References
- ^ Ziegler, Günter M. (1995), Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152, Berlin, New York: Springer-Verlag
- ^ Maclagan, Diane; Sturmfels, Bernd (2015). Introduction to Tropical Geometry. American Mathematical Soc. ISBN 9780821851982.
- ^ Mora, Teo; Robbiano, Lorenzo (1988). "The Gröbner fan of an ideal". Journal of Symbolic Computation. 6 (2–3): 183–208. doi:10.1016/S0747-7171(88)80042-7.
- ^ Bayer, David; Morrison, Ian (1988). "Standard bases and geometric invariant theory I. Initial ideals and state polytopes". Journal of Symbolic Computation. 6 (2–3): 209–217. doi:10.1016/S0747-7171(88)80043-9.
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