Polyhedral complex

Math concept

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 K {\displaystyle {\mathcal {K}}} is a set of polyhedra that satisfies the following conditions:

1. Every face of a polyhedron from K {\displaystyle {\mathcal {K}}} is also in K {\displaystyle {\mathcal {K}}} .
2. The intersection of any two polyhedra σ 1 , σ 2 ∈ K {\displaystyle \sigma _{1},\sigma _{2}\in {\mathcal {K}}} is a face of both σ 1 {\displaystyle \sigma _{1}} and σ 2 {\displaystyle \sigma _{2}} .

Note that the empty set is a face of every polyhedron, and so the intersection of two polyhedra in K {\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.

References

  1. ^ Ziegler, Günter M. (1995), Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152, Berlin, New York: Springer-Verlag
  2. ^ Maclagan, Diane; Sturmfels, Bernd (2015). Introduction to Tropical Geometry. American Mathematical Soc. ISBN 9780821851982.
  3. ^ Mora, Teo; Robbiano, Lorenzo (1988). "The Gröbner fan of an ideal". Journal of Symbolic Computation. 6 (2–3): 183–208. doi:10.1016/S0747-7171(88)80042-7.
  4. ^ Bayer, David; Morrison, Ian (1988). "Standard bases and geometric invariant theory I. Initial ideals and state polytopes". Journal of Symbolic Computation. 6 (2–3): 209–217. doi:10.1016/S0747-7171(88)80043-9.