Tu banner alternativo

Polyhedral complex

In the wide and varied world of Polyhedral complex, there are endless aspects to explore and discover. Whether it's Polyhedral complex itself, its impacts on society, or its evolution over time, there is always something new to learn and reflect on. From its origins to its influence today, Polyhedral complex remains a topic of constant interest and debate. In this article, we will dive into different aspects of Polyhedral complex, exploring its importance, its challenges, and its possible future scenarios. Join us on this exploration and let's discover more about Polyhedral complex together.

Tu banner alternativo

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

  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.