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Convex polytope

Index Convex polytope

A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn. [1]

70 relations: Aequationes Mathematicae, Affine space, Angle, Ball (mathematics), Big O notation, Bounded set, Branko Grünbaum, Compact space, Cone, Conical combination, Convex combination, Convex cone, Convex hull, Convex hull algorithms, Convex polygon, Convex set, Cylinder, Decision tree model, Discrete geometry, Edge (geometry), Elliptic geometry, Euclidean vector, Euler characteristic, Eulerian poset, Extreme point, Face (geometry), Facet (geometry), Fundamental group, Günter M. Ziegler, Glen Bredon, Graph isomorphism problem, Graphs and Combinatorics, Half-space (geometry), Hassler Whitney, Homeomorphism, Intersection (set theory), Isomorphism, Journal of Combinatorial Theory, Journal of the ACM, Lattice (order), Line (geometry), Linear combination, Linear independence, Linear inequality, Linear programming, Manifold, Mathematics, Matrix (mathematics), Micha Perles, N-sphere, ..., Nef polygon, Oriented matroid, Polygon, Polygonal chain, Polyhedral graph, Polyhedron, Polytope, Prism (geometry), Simple polytope, Simplex, Simplicial complex, Spherical polyhedron, Tessellation, Time complexity, Unique sink orientation, Upper and lower bounds, Vector space, Vertex (geometry), Vertex enumeration problem, Victor Klee. Expand index (20 more) »

Aequationes Mathematicae

Aequationes Mathematicae is a mathematical journal.

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Affine space

In mathematics, an affine space is a geometric structure that generalizes the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments.

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In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.

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Ball (mathematics)

In mathematics, a ball is the space bounded by a sphere.

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Big O notation

Big O notation is a mathematical notation that describes the limiting behaviour of a function when the argument tends towards a particular value or infinity.

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Bounded set

In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size.

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Branko Grünbaum

Branko Grünbaum (ברנקו גרונבאום; born 2 October 1929) is a Yugoslavian-born mathematician and a professor emeritus at the University of Washington in Seattle.

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Compact space

In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).

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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.

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Conical combination

Given a finite number of vectors x_1, x_2, \dots, x_n in a real vector space, a conical combination, conical sum, or weighted sumConvex Analysis and Minimization Algorithms by Jean-Baptiste Hiriart-Urruty, Claude Lemaréchal, 1993,, Mathematical Programming, by Melvyn W. Jeter (1986), of these vectors is a vector of the form where \alpha_i are non-negative real numbers.

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Convex combination

In convex geometry, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1.

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Convex cone

In linear algebra, a convex cone is a subset of a vector space over an ordered field that is closed under linear combinations with positive coefficients.

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Convex hull

In mathematics, the convex hull or convex envelope or convex closure of a set X of points in the Euclidean plane or in a Euclidean space (or, more generally, in an affine space over the reals) is the smallest convex set that contains X. For instance, when X is a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around X., p. 3.

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Convex hull algorithms

Algorithms that construct convex hulls of various objects have a broad range of applications in mathematics and computer science.

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Convex polygon

A convex polygon is a simple polygon (not self-intersecting) in which no line segment between two points on the boundary ever goes outside the polygon.

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Convex set

In convex geometry, a convex set is a subset of an affine space that is closed under convex combinations.

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A cylinder (from Greek κύλινδρος – kulindros, "roller, tumbler"), has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes.

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Decision tree model

In computational complexity and communication complexity theories the decision tree model is the model of computation or communication in which an algorithm or communication process is considered to be basically a decision tree, i.e., a sequence of branching operations based on comparisons of some quantities, the comparisons being assigned the unit computational cost.

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Discrete geometry

Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects.

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Edge (geometry)

In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope.

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Elliptic geometry

Elliptic geometry is a geometry in which Euclid's parallel postulate does not hold.

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Euclidean vector

In mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric or spatial vector, or—as here—simply a vector) is a geometric object that has magnitude (or length) and direction.

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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 number that describes a topological space's shape or structure regardless of the way it is bent.

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Eulerian poset

In combinatorial mathematics, an Eulerian poset is a graded poset in which every nontrivial interval has the same number of elements of even rank as of odd rank.

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Extreme point

In mathematics, an extreme point of a convex set S in a real vector space is a point in S which does not lie in any open line segment joining two points of S. Intuitively, an extreme point is a "vertex" of S.

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Face (geometry)

In solid geometry, a face is a flat (planar) surface that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by flat faces is a polyhedron.

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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.

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Fundamental group

In the mathematical field of algebraic topology, the fundamental group is a mathematical group associated to any given pointed topological space that provides a way to determine when two paths, starting and ending at a fixed base point, can be continuously deformed into each other.

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Günter M. Ziegler

Günter Matthias Ziegler (born 19 May 1963) is a German mathematician.

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Glen Bredon

Glen Eugene Bredon (August 24, 1932 in Fresno, California – May 8, 2000) was an American mathematician who worked in the area of topology.

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Graph isomorphism problem

The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic.

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Graphs and Combinatorics

Graphs and Combinatorics (ISSN 0911-0119, abbreviated Graphs Combin.) is a peer-reviewed academic journal in graph theory and combinatorics published by Springer Japan.

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Half-space (geometry)

In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional Euclidean space.

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Hassler Whitney

Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician.

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In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function.

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Intersection (set theory)

In mathematics, the intersection A ∩ B of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements.

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In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.

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Journal of Combinatorial Theory

The Journal of Combinatorial Theory, Series A and Series B, are mathematical journals specializing in combinatorics and related areas.

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Journal of the ACM

The Journal of the ACM is a peer-reviewed scientific journal covering computer science in general, especially theoretical aspects.

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Lattice (order)

A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.

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Line (geometry)

The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth.

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Linear combination

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants).

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Linear independence

In the theory of vector spaces, a set of vectors is said to be if one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said to be.

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Linear inequality

In mathematics a linear inequality is an inequality which involves a linear function.

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Linear programming

Linear programming (LP, also called linear optimization) is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships.

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In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

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Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

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Matrix (mathematics)

In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

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Micha Perles

Micha Asher Perles is an Israeli mathematician working in geometry, a professor emeritus at the Hebrew University.

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In mathematics, the n-sphere is the generalization of the ordinary sphere to spaces of arbitrary dimension.

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Nef polygon

Nef polygons and Nef polyhedra are the sets of polygons (resp. polyhedra) which can be obtained from a finite set of halfplanes (halfspaces) by Boolean operations of set intersection and set complement.

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Oriented matroid

An oriented matroid is a mathematical structure that abstracts the properties of directed graphs and of arrangements of vectors in a vector space over an ordered field (particularly for partially ordered vector spaces).

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In elementary geometry, a polygon is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit.

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Polygonal chain

In geometry, a polygonal chain is a connected series of line segments.

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Polyhedral graph

In geometric graph theory, a branch of mathematics, a polyhedral graph is the undirected graph formed from the vertices and edges of a convex polyhedron.

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In geometry, a polyhedron (plural polyhedra or polyhedrons) is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices.

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In elementary geometry, a polytope is a geometric object with "flat" sides.

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Prism (geometry)

In geometry, a prism is a polyhedron comprising an n-sided polygonal base, a second base which is a translated copy (rigidly moved without rotation) of the first, and n other faces (necessarily all parallelograms) joining corresponding sides of the two bases.

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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).

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In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions.

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Simplicial complex

In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration).

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Spherical polyhedron

In mathematics, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons.

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A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps.

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Time complexity

In computer science, the time complexity is the computational complexity that describes the amount of time it takes to run an algorithm.

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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 vertex for which all adjoining edges are oriented inward (i.e. towards that vertex).

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Upper and lower bounds

In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set (K, ≤) is an element of K which is greater than or equal to every element of S. The term lower bound is defined dually as an element of K which is less than or equal to every element of S. A set with an upper bound is said to be bounded from above by that bound, a set with a lower bound is said to be bounded from below by that bound.

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Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

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Vertex (geometry)

In geometry, a vertex (plural: vertices or vertexes) is a point where two or more curves, lines, or edges meet.

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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 object's vertices given some formal representation of the object.

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Victor Klee

Victor L. Klee, Jr. (September 18, 1925, San Francisco – August 17, 2007, Lakewood, Ohio) was a mathematician specialising in convex sets, functional analysis, analysis of algorithms, optimization, and combinatorics.

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Combinatorial isomorphism, Convex polyhedra, Convex polyhedron, Convex polytopes, Face lattice, Facet enumeration, Facet enumeration problem, H-description, Halfspace representation, Nonconvex, Polytopal graph, Polytope graph, V-description.


[1] https://en.wikipedia.org/wiki/Convex_polytope

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