From Wikipedia, the free encyclopedia
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 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
Fans
A fan is a polyhedral complex in which every polyhedron is a
cone from the origin. Examples of fans include:
References
-
^ Ziegler, Günter M. (1995), Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152, Berlin, New York:
Springer-Verlag
-
^
Maclagan, Diane; Sturmfels, Bernd (2015).
Introduction to Tropical Geometry. American Mathematical Soc.
ISBN
9780821851982.
-
^ 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.
-
^ 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.