Exposed Point

picture info	Exposed Point In mathematics, an exposed point of a convex set C is a point x\in C at which some continuous linear functional attains its strict maximum over C. Such a functional is then said to ''expose'' x. There can be many exposing functionals for x. The set of exposed points of C is usually denoted \exp(C). A stronger notion is that of ''strongly exposed point'' of C which is an exposed point x \in C such that some exposing functional f of x attains its strong maximum over C at x, i.e. for each sequence (x_n) \subset C we have the following implication: f(x_n) \to \max f(C) \Longrightarrow \, x_n -x\, \to 0. The set of all strongly exposed points of C is usually denoted \operatorname\exp(C). There are two weaker notions, that of extreme point In mathematics, an extreme point of a convex set S in a real or complex vector space is a point in S which does not lie in any open line segment joining two points of S. In linear programming problems, an extreme point is also called vertex or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Convex Set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex region is a subset that intersects every line into a single line segment (possibly empty). For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex. The boundary of a convex set is always a convex curve. The intersection of all the convex sets that contain a given subset of Euclidean space is called the convex hull of . It is the smallest convex set containing . A convex function is a real-valued function defined on an interval with the property that its epigraph (the set of points on or above the graph of the function) is a convex set. Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Continuous Linear Functional In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded linear operator if and only if it is a continuous linear operator. Continuous linear operators Characterizations of continuity Suppose that F : X \to Y is a linear operator between two topological vector spaces (TVSs). The following are equivalent: F is continuous. F is continuous at some point x \in X. F is continuous at the origin in X. if Y is locally convex then this list may be extended to include: for every continuous seminorm q on Y, there exists a continuous seminorm p on X such that q \circ F \leq p. if X and Y are both Hausdorff locally convex spaces then this list may be extended to include: F is weakly continuous and its transpose ^t F : Y^ \to X^ maps equicontinuous subsets of Y^ to equicontinuous subsets o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Maximum In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the ''local'' or ''relative'' extrema), or on the entire domain (the ''global'' or ''absolute'' extrema). Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions. As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum. Definition A real-valued function ''f'' defined on a domain ''X'' has a global (or absolute) maximum point at ''x''∗, if for all ''x'' in ''X''. Similarly, the function has a global (or absolute) minimum point at ''x''∗, if for all ''x'' in ''X''. The value of the function a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Extreme Point In mathematics, an extreme point of a convex set S in a real or complex vector space is a point in S which does not lie in any open line segment joining two points of S. In linear programming problems, an extreme point is also called vertex or corner point of S. Definition Throughout, it is assumed that X is a real or complex vector space. For any p, x, y \in X, say that p x and y if x \neq y and there exists a 0 < t < 1 such that $p = t x + (1-t) y.$ If $K$ is a subset of $X$ and $p \in K,$ then $p$ is called an of $K$ if it does not lie between any two points of $K.$ That is, if there does exist $x, y \in K$ and $0 < t < 1$ such that $x \neq y$ and $p = t x + (1-t) y.$ The set of all extreme points of $K$ is denoted by $\operatorname(K).$ Generalizations If $S$ is a subset of a vector space then a linear ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Support Point Support may refer to: Arts, entertainment, and media * Supporting character Business and finance * Support (technical analysis) * Child support * Customer support * Income Support Construction * Support (structure), or lateral support, a type of structural support to help prevent sideways movement * Structural support, architectural components that include arches, beams, columns, balconies, and stretchers Law and politics * Advocacy, in politics, support for constituencies, issues, or legislation * Lateral and subjacent support, a legal term Mathematics Mathematics (generally) * Support (mathematics), subset of the domain of a function where it is non-zero valued * Support (measure theory), a subset of a measurable space * Supporting hyperplane, sometimes referred to as support Statistics * Support, the natural logarithm of the likelihood ratio, as used in phylogenetics * Method of support, in statistics, a technique that is used to make inferences from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Mathematical Analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were i ... [...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 co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]