Polyhedral complex
# Polyhedral complex

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.

## Definition

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.

## Examples

* 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.

## 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.
* 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.
