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

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A polyhedral complex   is a set of polyhedra that satisfies the following conditions:

1. Every face of a polyhedron from   is also in  .
2. The intersection of any two polyhedra   is a face of both   and  .

Note that the empty set is a face of every polyhedron, and so the intersection of two polyhedra in   may be empty.

Examples

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Fans

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A fan is a polyhedral complex in which every polyhedron is a cone from the origin. Examples of fans include:

References

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  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.