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.cite-ref-1[1] Polyhedral complexes generalize simplicial complexes and arise in various areas of polyhedral geometry, such as tropical geometry, splines and hyperplane arrangements.
Contents
• Examples
• Fans
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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 σ 1 , σ σ 2 ∈ ∈ K {\displaystyle \sigma _{1},\sigma _{2}\in {\mathcal {K}}} is a face of both σ σ 1 {\displaystyle \sigma _{1}} and σ 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.cite-ref-maclagan-2-0[2]
• Simplicial complexes are polyhedral complexes in which every polyhedron is a simplex.
• 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.
• A tropical variety obtained by tropicalizing an algebraic variety over a valued field with trivial valuation.
• The recession fan of a tropical variety.
References
cite-note-11. ↑ citerefziegler1995Ziegler, Günter M. (1995), Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152, Berlin, New York: Springer-Verlag
cite-note-maclagan-22. ↑ citerefmaclagansturmfels2015Maclagan, Diane; Sturmfels, Bernd (2015). Introduction to Tropical Geometry. American Mathematical Soc. ISBN 9780821851982.