Polyhedral complex

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.

• 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.

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.