polyhedral complex

A polyhedral complex $\backslash mathcal\{K\}$ is a set of polyhedra that satisfies the following conditions:

:1. Every face of a polyhedron from $\backslash mathcal\{K\}$ is also in $\backslash mathcal\{K\}$.

:2. The intersection of any two polyhedra $\backslash sigma\_1,\; \backslash sigma\_2\; \backslash in\; \backslash mathcal\{K\}$ is a face of both $\backslash sigma\_1$ and $\backslash sigma\_2$.

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

• Tropical varieties are polyhedral complexes satisfying a certain balancing condition.{{cite book|last=Maclagan|first=Diane|authorlink=Diane Maclagan|last2=Sturmfels|first2=Bernd |title= Introduction to Tropical Geometry |title-link= Introduction to Tropical Geometry |year=2015|publisher=American Mathematical Soc.|isbn=9780821851982 }}
• Simplicial complexes are polyhedral complexes in which every polyhedron is a simplex.
• Voronoi diagrams.
• Splines.

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.{{Cite journal|title=The Gröbner fan of an ideal |language=en|doi=10.1016/S0747-7171(88)80042-7|volume=6|issue=2–3 |journal=Journal of Symbolic Computation|pages=183–208 | last2 = Robbiano | first2 = Lorenzo | last1 = Mora | first1 = Teo|year=1988 |doi-access=free }}{{Cite journal|title=Standard bases and geometric invariant theory I. Initial ideals and state polytopes|language=en|doi=10.1016/S0747-7171(88)80043-9|volume=6|issue=2–3|journal=Journal of Symbolic Computation|pages=209–217 | last1 = Bayer | first1 = David | last2 = Morrison | first2 = Ian|year=1988|doi-access=free}}
• 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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Category:Polyhedra