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 (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.