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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 ${\displaystyle {\mathcal {K}}}$ 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.
• 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.

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.

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