|
Carathéodory's Theorem (convex Hull)
Carathéodory's theorem is a theorem in convex geometry. It states that if a point x lies in the convex hull \mathrm(P) of a set P\subset \R^d, then x can be written as the convex combination of at most d+1 points in P. More sharply, x can be written as the convex combination of at most d+1 ''extremal'' points in P, as non-extremal points can be removed from P without changing the membership of ''x'' in the convex hull. Its equivalent theorem for conical combinations states that if a point x lies in the conical hull \mathrm(P) of a set P\subset \R^d, then x can be written as the conical combination of at most d points in P. The similar theorems of Helly and Radon are closely related to Carathéodory's theorem: the latter theorem can be used to prove the former theorems and vice versa. The result is named for Constantin Carathéodory, who proved the theorem in 1911 for the case when P is compact. In 1914 Ernst Steinitz expanded Carathéodory's theorem for arbitrary set. Exa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Geometry
In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbers, integral geometry, linear programming, probability theory, game theory, etc. Classification According to the Mathematics Subject Classification MSC2010, the mathematical discipline ''Convex and Discrete Geometry'' includes three major branches: * general convexity * polytopes and polyhedra * discrete geometry (though only portions of the latter two are included in convex geometry). General convexity is further subdivided as follows: *axiomatic and generalized convexity *convex sets without dimension restrictions *convex sets in topological vector spaces *convex sets in 2 dimensions (including convex curves) *convex sets in 3 dimensions (including convex surfaces) *convex sets in ''n'' dimensions (including convex hy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PPAD (complexity)
In computer science, PPAD ("Polynomial Parity Arguments on Directed graphs") is a complexity class introduced by Christos Papadimitriou in 1994. PPAD is a subclass of TFNP based on functions that can be shown to be total by a parity argument. The class attracted significant attention in the field of algorithmic game theory because it contains the problem of computing a Nash equilibrium: this problem was shown to be complete for PPAD by Daskalakis, Goldberg and Papadimitriou with at least 3 players and later extended by Chen and Deng to 2 players.*. Definition PPAD is a subset of the class TFNP, the class of function problems in FNP that are guaranteed to be total. The TFNP formal definition is given as follows: :A binary relation P(''x'',''y'') is in TFNP if and only if there is a deterministic polynomial time algorithm that can determine whether P(''x'',''y'') holds given both ''x'' and ''y'', and for every ''x'', there exists a ''y'' such that P(''x'',''y'') holds. Subclasses ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Transversal Theory
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Hulls
Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope, a polytope with a convex set of points ** Convex metric space, a generalization of the convexity notion in abstract metric spaces * Convex function, when the line segment between any two points on the graph of the function lies above or on the graph * Convex conjugate, of a function * Convexity (algebraic geometry), a restrictive technical condition for algebraic varieties originally introduced to analyze Kontsevich moduli spaces Economics and finance * Convexity (finance), second derivatives in financial modeling generally * Convexity in economics * Bond convexity, a measure of the sensitivity of the duration of a bond to changes in interest rates * Convex preferences, an individual's ordering of various outcomes Other uses * Convex Com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Articles Containing Proofs
Article often refers to: * Article (grammar), a grammatical element used to indicate definiteness or indefiniteness * Article (publishing), a piece of nonfictional prose that is an independent part of a publication Article may also refer to: Government and law * Article (European Union), articles of treaties of the European Union * Articles of association, the regulations governing a company, used in India, the UK and other countries * Articles of clerkship, the contract accepted to become an articled clerk * Articles of Confederation, the predecessor to the current United States Constitution *Article of Impeachment, a formal document and charge used for impeachment in the United States * Articles of incorporation, for corporations, U.S. equivalent of articles of association * Articles of organization, for limited liability organizations, a U.S. equivalent of articles of association Other uses * Article, an HTML element, delimited by the tags and * Article of clothing, an ite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PlanetMath
PlanetMath is a free, collaborative, mathematics online encyclopedia. The emphasis is on rigour, openness, pedagogy, real-time content, interlinked content, and also community of about 24,000 people with various maths interests. Intended to be comprehensive, the project is currently hosted by the University of Waterloo. The site is owned by a US-based nonprofit corporation, "PlanetMath.org, Ltd". PlanetMath was started when the popular free online mathematics encyclopedia MathWorld was temporarily taken offline for 12 months by a court injunction as a result of the CRC Press lawsuit against the Wolfram Research company and its employee (and MathWorld's author) Eric Weisstein. Materials The main PlanetMath focus is on encyclopedic entries. It formerly operated a self-hosted forum, but now encourages discussion via Gitter. , the encyclopedia hosted about 9,289 entries and over 16,258 ''concepts'' (a concept may be for example a specific notion defined within a more general entry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Choquet Theory
In mathematics, Choquet theory, named after Gustave Choquet, is an area of functional analysis and convex analysis concerned with measures which have support on the extreme points of a convex set ''C''. Roughly speaking, every vector of ''C'' should appear as a weighted average of extreme points, a concept made more precise by generalizing the notion of weighted average from a convex combination to an integral taken over the set ''E'' of extreme points. Here ''C'' is a subset of a real vector space ''V'', and the main thrust of the theory is to treat the cases where ''V'' is an infinite-dimensional (locally convex Hausdorff) topological vector space along lines similar to the finite-dimensional case. The main concerns of Gustave Choquet were in potential theory. Choquet theory has become a general paradigm, particularly for treating convex cones as determined by their extreme rays, and so for many different notions of ''positivity'' in mathematics. The two ends of a line segment dete ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Krein–Milman Theorem
In the mathematical theory of functional analysis, the Krein–Milman theorem is a proposition about compact convex sets in locally convex topological vector spaces (TVSs). This theorem generalizes to infinite-dimensional spaces and to arbitrary compact convex sets the following basic observation: a convex (i.e. "filled") triangle, including its perimeter and the area "inside of it", is equal to the convex hull of its three vertices, where these vertices are exactly the extreme points of this shape. This observation also holds for any other convex polygon in the plane \R^2. Statement and definitions Preliminaries and definitions Throughout, X will be a real or complex vector space. For any elements x and y in a vector space, the set , y:= \ is called the or closed interval between x and y. The or open interval between x and y is (x, x) := \varnothing when x = y while it is (x, y) := \ when x \neq y; it satisfies (x, y) = , y\setminus \ and , y= (x, y) \cup \. The points x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tverberg's Theorem
In discrete geometry, Tverberg's theorem, first stated by , is the result that sufficiently many points in ''d''-dimensional Euclidean space can be partitioned into subsets with intersecting convex hulls. Specifically, for any set of :(d + 1)(r - 1) + 1\ points there exists a point ''x'' (not necessarily one of the given points) and a partition of the given points into ''r'' subsets, such that ''x'' belongs to the convex hull of all of the subsets. The partition resulting from this theorem is known as a Tverberg partition. Examples For ''r'' = 2, Tverberg's theorem states that any ''d'' + 2 points may be partitioned into two subsets with intersecting convex hulls; this special case is known as Radon's theorem. In this case, for points in general position, there is a unique partition. The case ''r'' = 3 and ''d'' = 2 states that any seven points in the plane may be partitioned into three subsets with intersecting convex hulls. The illustrat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kirchberger's Theorem
Kirchberger's theorem is a theorem in discrete geometry, on linear separability. The two-dimensional version of the theorem states that, if a finite set of red and blue points in the Euclidean plane has the property that, for every four points, there exists a line separating the red and blue points within those four, then there exists a single line separating all the red points from all the blue points. Donald Watson phrases this result more colorfully, with a farmyard analogy: More generally, for finitely many red and blue points in d-dimensional Euclidean space, if the red and blue points in every subset of d+2 of the points are linearly separable, then all the red points and all the blue points are linearly separable. Another equivalent way of stating the result is that, if the convex hulls of finitely many red and blue points have a nonempty intersection, then there exists a subset of d+2 points for which the convex hulls of the red and blue points in the subsets also intersect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shapley–Folkman Lemma
The Shapley–Folkman lemma is a result in convex geometry that describes the Minkowski addition of sets in a vector space. It is named after mathematicians Lloyd Shapley and Jon Folkman, but was first published by the economist Ross M. Starr. The lemma may be intuitively understood as saying that, if the number of summed sets exceeds the dimension of the vector space, then their Minkowski sum is approximately convex. Related results provide more refined statements about ''how close'' the approximation is. For example, the Shapley–Folkman theorem provides an upper bound on the distance between any point in the Minkowski sum and its convex hull. This upper bound is sharpened by the Shapley–Folkman–Starr theorem (alternatively, Starr's corollary). The Shapley–Folkman lemma has applications in economics, optimization and probability theory. In economics, it can be used to extend results proved for convex preferences to non-convex preferences. In op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PLS (complexity)
In computational complexity theory, Polynomial Local Search (PLS) is a complexity class that models the difficulty of finding a locally optimal solution to an optimization problem. The main characteristics of problems that lie in PLS are that the cost of a solution can be calculated in polynomial time and the neighborhood of a solution can be searched in polynomial time. Therefore it is possible to verify whether or not a solution is a local optimum in polynomial time. Furthermore, depending on the problem and the algorithm that is used for solving the problem, it might be faster to find a local optimum instead of a global optimum. Description When searching for a local optimum, there are two interesting issues to deal with: First how to find a local optimum, and second how long it takes to find a local optimum. For many local search algorithms, it is not known, whether they can find a local optimum in polynomial time or not. So to answer the question of how long it takes to find a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
