# Category: Polyhedral combinatorics

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 constr
Extension complexity
In convex geometry and polyhedral combinatorics, the extension complexity is a convex polytope is the smallest number of facets among convex polytopes that have as a projection. In this context, is ca
Reye configuration
In geometry, the Reye configuration, introduced by Theodor Reye, is a configuration of 12 points and 16 lines.Each point of the configuration belongs to four lines, and each line contains three points
Lectures in Geometric Combinatorics
Lectures in Geometric Combinatorics is a textbook on polyhedral combinatorics. It was written by Rekha R. Thomas, based on a course given by Thomas at the 2004 Park City Mathematics Institute, and pub
Hirsch conjecture
In mathematical programming and polyhedral combinatorics, the Hirsch conjecture is the statement that the edge-vertex graph of an n-facet polytope in d-dimensional Euclidean space has diameter no more
H-vector
In algebraic combinatorics, the h-vector of a simplicial polytope is a fundamental invariant of the polytope which encodes the number of faces of different dimensions and allows one to express the Deh
Neighborly polytope
In geometry and polyhedral combinatorics, a k-neighborly polytope is a convex polytope in which every set of k or fewer vertices forms a face. For instance, a 2-neighborly polytope is a polytope in wh
Perles configuration
In geometry, the Perles configuration is a system of nine points and nine lines in the Euclidean plane for which every combinatorially equivalent realization has at least one irrational number as one
Facet (geometry)
In geometry, a facet is a feature of a polyhedron, polytope, or related geometric structure, generally of dimension one less than the structure itself. More specifically: * In three-dimensional geome
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a num
Polyhedral combinatorics
Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimension
Unique sink orientation
In mathematics, a unique sink orientation is an orientation of the edges of a polytope such that, in every face of the polytope (including the whole polytope as one of the faces), there is exactly one
Euler's Gem
Euler's Gem: The Polyhedron Formula and the Birth of Topology is a book on the formula for the Euler characteristic of convex polyhedra and its connections to the history of topology. It was written b
Vertex enumeration problem
In mathematics, the vertex enumeration problem for a polytope, a polyhedral cell complex, a hyperplane arrangement, or some other object of discrete geometry, is the problem of determination of the ob
Kalai's 3^d conjecture
In geometry, Kalai's 3d conjecture is a conjecture on the polyhedral combinatorics of centrally symmetric polytopes, made by Gil Kalai in 1989. It states that every d-dimensional centrally symmetric p
Balinski's theorem
In polyhedral combinatorics, a branch of mathematics, Balinski's theorem is a statement about the graph-theoretic structure of three-dimensional convex polyhedra and higher-dimensional convex polytope
Upper bound theorem
In mathematics, the upper bound theorem states that cyclic polytopes have the largest possible number of faces among all convex polytopes with a given dimension and number of vertices. It is one of th
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
Eberhard's theorem
In mathematics, and more particularly in polyhedral combinatorics, Eberhard's theorem partially characterizes the multisets of polygons that can form the faces of simple convex polyhedra. It states th
Gale diagram
In the mathematical discipline of polyhedral combinatorics, the Gale transform turns the vertices of any convex polytope into a set of vectors or points in a space of a different dimension, the Gale d
Dehn–Sommerville equations
In mathematics, the Dehn–Sommerville equations are a complete set of linear relations between the numbers of faces of different dimension of a simplicial polytope. For polytopes of dimension 4 and 5,
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
Stable matching polytope
In mathematics, economics, and computer science, the stable matching polytope or stable marriage polytope is a convex polytope derived from the solutions to an instance of the stable matching problem.
Cyclic polytope
In mathematics, a cyclic polytope, denoted C(n,d), is a convex polytope formed as a convex hull of n distinct points on a rational normal curve in Rd, where n is greater than d. These polytopes were s