# Polyhedral complex

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Polyhedral_complex
> Markdown URL: https://mediated.wiki/source/Polyhedral_complex.md
> Source: https://en.wikipedia.org/wiki/Polyhedral_complex
> Source revision: 1339171799
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

{{short description|Math concept}}
In [mathematics](/source/mathematics), a '''polyhedral complex''' is a set of [polyhedra](/source/Polyhedron) in a [real](/source/Real_number) [vector space](/source/vector_space) that fit together in a specific way.<ref>{{Citation | last=Ziegler | first=Günter M. | title=Lectures on Polytopes | publisher=[Springer-Verlag](/source/Springer-Verlag) | location=Berlin, New York | series=Graduate Texts in Mathematics | year=1995 | volume=152}}</ref>  Polyhedral complexes generalize [simplicial complexes](/source/Simplicial_complex) and arise in various areas of polyhedral geometry, such as [tropical geometry](/source/tropical_geometry), [splines](/source/spline_(mathematics)) and [hyperplane arrangements](/source/hyperplane_arrangements).

==Definition==
A '''polyhedral complex''' <math>\mathcal{K}</math> is a set of [polyhedra](/source/Polyhedron) that satisfies the following conditions:
:1. Every [face](/source/Face_(geometry)) of a polyhedron from <math>\mathcal{K}</math> is also in <math>\mathcal{K}</math>.
:2. The [intersection](/source/Set_intersection) of any two polyhedra <math>\sigma_1, \sigma_2 \in \mathcal{K}</math> is a face of both <math>\sigma_1</math> and <math>\sigma_2</math>.
Note that the empty set is a face of every polyhedron, and so the intersection of two polyhedra in <math>\mathcal{K}</math> may be empty.

==Examples==
* [Tropical varieties](/source/Tropical_geometry) are polyhedral complexes satisfying a certain ''balancing condition''.<ref name=Maclagan>{{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 }}</ref>
* [Simplicial complexes](/source/Simplicial_complex) are polyhedral complexes in which every polyhedron is a [simplex](/source/simplex).
* [Voronoi diagrams](/source/Voronoi_diagrams).
* [Splines](/source/Spline_(mathematics)).

==Fans==
A '''(polyhedral) fan''' is a polyhedral complex in which every polyhedron is a [cone](/source/cone) from the origin.  Examples of fans include:
* The [normal fan](/source/normal_fan) of a [polytope](/source/polytope).
* The fan associated to a toric variety (see {{section link|Toric variety#Fundamental theorem for toric geometry}}).
* The [Gröbner fan](/source/Gr%C3%B6bner_fan) of an [ideal](/source/Ideal_(ring_theory)) of a [polynomial ring](/source/polynomial_ring).<ref>{{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 }}</ref><ref>{{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}}</ref>
* A tropical variety obtained by tropicalizing an [algebraic variety](/source/algebraic_variety) over a [valued field](/source/Valuation_(algebra)) with trivial valuation.
* The ''recession fan'' of a tropical variety.

== References ==
{{Reflist}}

{{Topology}}

Category:Polyhedra

---
Adapted from the Wikipedia article [Polyhedral complex](https://en.wikipedia.org/wiki/Polyhedral_complex) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Polyhedral_complex?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.