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Polyhedral complex

Math concept

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Source updated February 19, 2026

In mathematics, a polyhedral complex is a set of polyhedra in a real vector space that fit together in a specific way. Polyhedral complexes generalize simplicial complexes and arise in various areas of polyhedral geometry, such as tropical geometry, splines and hyperplane arrangements.

01Definition

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

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

2. The intersection of any two polyhedra $\sigma _{1},\sigma _{2}\in {\mathcal {K}}$ is a face of both $\sigma _{1}$ and $\sigma _{2}$ .

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

02Examples

Tropical varieties are polyhedral complexes satisfying a certain balancing condition.
Simplicial complexes are polyhedral complexes in which every polyhedron is a simplex.
Voronoi diagrams.
Splines.

03Fans

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 Toric variety § Fundamental theorem for toric geometry).
The Gröbner fan of an ideal of a polynomial ring.
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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Sources and credits

This article is adapted from the Wikipedia article “Polyhedral complex”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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