{{short description|Math concept}} In mathematics, a '''polyhedral complex''' is a set of polyhedra in a real vector space that fit together in a specific way.<ref>{{Citation | last=Ziegler | first=Günter M. | title=Lectures on Polytopes | publisher=Springer-Verlag | location=Berlin, New York | series=Graduate Texts in Mathematics | year=1995 | volume=152}}</ref> 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''' <math>\mathcal{K}</math> is a set of polyhedra that satisfies the following conditions: :1. Every face of a polyhedron from <math>\mathcal{K}</math> is also in <math>\mathcal{K}</math>. :2. The 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 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 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 {{section link|Toric variety#Fundamental theorem for toric geometry}}). * The Gröbner fan of an ideal of a 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 over a valued field with trivial valuation. * The ''recession fan'' of a tropical variety.
== References == {{Reflist}}
{{Topology}}
Category:Polyhedra