Polyhedral complex

In mathematics, a polyhedral complex is a set of polyhedra in a real vector space that fit together in a specific way.^[1] 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 ${\displaystyle {\mathcal {K}}}$ is a set of polyhedra that satisfies the following conditions:

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

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

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

Examples

• Tropical varieties are polyhedral complexes satisfying a certain balancing condition.^[2]
• 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.^[3][4]
• A tropical variety obtained by tropicalizing an algebraic variety over a valued field with trivial valuation.
• The recession fan of a tropical variety.