Concept
Polyhedral combinatorics
Related concepts (16)
Steinitz's theorem
In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the 3-vertex-connected planar graphs. That is, every convex polyhedron forms a 3-connected planar graph, and every 3-connected planar graph can be represented as the graph of a convex polyhedron. For this reason, the 3-connected planar graphs are also known as polyhedral graphs.
Linear programming relaxation
In mathematics, the relaxation of a (mixed) integer linear program is the problem that arises by removing the integrality constraint of each variable. For example, in a 0–1 integer program, all constraints are of the form The relaxation of the original integer program instead uses a collection of linear constraints The resulting relaxation is a linear program, hence the name.
Birkhoff polytope
The Birkhoff polytope Bn (also called the assignment polytope, the polytope of doubly stochastic matrices, or the perfect matching polytope of the complete bipartite graph ) is the convex polytope in RN (where N = n2) whose points are the doubly stochastic matrices, i.e., the n × n matrices whose entries are non-negative real numbers and whose rows and columns each add up to 1. It is named after Garrett Birkhoff. The Birkhoff polytope has n! vertices, one for each permutation on n items.
Simple polytope
In geometry, a d-dimensional simple polytope is a d-dimensional polytope each of whose vertices are adjacent to exactly d edges (also d facets). The vertex figure of a simple d-polytope is a (d – 1)-simplex. Simple polytopes are topologically dual to simplicial polytopes. The family of polytopes which are both simple and simplicial are simplices or two-dimensional polygons. A simple polyhedron is a three-dimensional polyhedron whose vertices are adjacent to three edges and three faces.
Integral polytope
In geometry and polyhedral combinatorics, an integral polytope is a convex polytope whose vertices all have integer Cartesian coordinates. That is, it is a polytope that equals the convex hull of its integer points. Integral polytopes are also called lattice polytopes or Z-polytopes. The special cases of two- and three-dimensional integral polytopes may be called polygons or polyhedra instead of polytopes, respectively. An -dimensional regular simplex can be represented as an integer polytope in , the convex hull of the integer points for which one coordinate is one and the rest are zero.
Simplicial polytope
In geometry, a simplicial polytope is a polytope whose facets are all simplices. For example, a simplicial polyhedron in three dimensions contains only triangular faces and corresponds via Steinitz's theorem to a maximal planar graph. They are topologically dual to simple polytopes. Polytopes which are both simple and simplicial are either simplices or two-dimensional polygons. Simplicial polyhedra include: Bipyramids Gyroelongated dipyramids Deltahedra (equilateral triangles) Platonic tetrahedron, octahed
Simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example, a 0-dimensional simplex is a point, a 1-dimensional simplex is a line segment, a 2-dimensional simplex is a triangle, a 3-dimensional simplex is a tetrahedron, and a 4-dimensional simplex is a 5-cell. Specifically, a k-simplex is a k-dimensional polytope which is the convex hull of its k + 1 vertices.
Abstract polytope
In mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points and lines. A geometric polytope is said to be a realization of an abstract polytope in some real N-dimensional space, typically Euclidean. This abstract definition allows more general combinatorial structures than traditional definitions of a polytope, thus allowing new objects that have no counterpart in traditional theory.
Discrete geometry
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.
Convex polytope
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the -dimensional Euclidean space . Most texts use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others (including this article) allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue.