Polyhedral complex explained

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

l{K} is a set of polyhedra that satisfies the following conditions:

1. Every face of a polyhedron from

l{K} is also in l{K} .

2. The intersection of any two polyhedra

\sigma _{1,} \sigma _{2} \in l{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 l{K} may be empty.

Examples

• Tropical varieties are polyhedral complexes satisfying a certain balancing condition.^[1]
• 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.^{[2] [3]}
• A tropical variety obtained by tropicalizing an algebraic variety over a valued field with trivial valuation.
• The recession fan of a tropical variety.

Notes and References

1. Book: Maclagan, Diane. Diane Maclagan

   . Diane Maclagan. Sturmfels. Bernd . Introduction to Tropical Geometry . Introduction to Tropical Geometry . 2015. American Mathematical Soc.. 9780821851982 .

2. The Gröbner fan of an ideal . en. 10.1016/S0747-7171(88)80042-7. 6. 2–3 . Journal of Symbolic Computation. 183–208 . Robbiano . Lorenzo . Mora . Teo. 1988 . free .
3. Standard bases and geometric invariant theory I. Initial ideals and state polytopes. en. 10.1016/S0747-7171(88)80043-9. 6. 2–3. Journal of Symbolic Computation. 209–217 . Bayer . David . Morrison . Ian. 1988. free.