title: Polyhedral complex
text: # 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.
  