In mathematics, a polyhedral complex is a set of

polyhedra In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on ...

in a

real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...

that fit together in a specific way. Polyhedral complexes generalize

simplicial complexes In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial se ...

and arise in various areas of polyhedral geometry, such as

tropical geometry In mathematics, tropical geometry is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication is replaced with ordinary addition: : x \oplus y = \min\, : x \otimes y = x + y. So f ...

, splines and hyperplane arrangements.

Definition

A polyhedral complex

\mathcal

is a set of

that satisfies the following conditions: :1. Every

face The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may aff ...

of a polyhedron from

\mathcal

is also in

\mathcal

. :2. The intersection of any two polyhedra

\sigma_1, \sigma_2 \in \mathcal

is a face of both

\sigma_1

and

\sigma_2

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

\mathcal

may be empty.

Examples

* Tropical varieties are polyhedral complexes satisfying a certain ''balancing condition''. *

Simplicial complexes In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial se ...

are polyhedral complexes in which every polyhedron is a simplex. *

Voronoi diagrams In mathematics, a Voronoi diagram is a Partition of a set, partition of a plane (geometry), plane into regions close to each of a given set of objects. In the simplest case, these objects are just finitely many points in the plane (called seeds, s ...

. * Splines.

Fans

A fan is a polyhedral complex in which every polyhedron is a

cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines con ...

from the origin. Examples of fans include: * The normal fan of a

polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...

. * The Gröbner fan of an

ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...

of a

polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...

. * A tropical variety obtained by tropicalizing an

algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...

over a

valued field Value or values may refer to: Ethics and social * Value (ethics) wherein said concept may be construed as treating actions themselves as abstract objects, associating value to them ** Values (Western philosophy) expands the notion of value beyo ...

with trivial valuation. * The ''recession fan'' of a tropical variety.

References

{{Topology Polyhedra