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Supporting Hyperplane
In geometry, a supporting hyperplane of a set S in Euclidean space \mathbb R^n is a hyperplane that has both of the following two properties: * S is entirely contained in one of the two closed half-spaces bounded by the hyperplane, * S has at least one boundary-point on the hyperplane. Here, a closed half-space is the half-space that includes the points within the hyperplane. Supporting hyperplane theorem This theorem states that if S is a convex set in the topological vector space X=\mathbb^n, and x_0 is a point on the boundary of S, then there exists a supporting hyperplane containing x_0. If x^* \in X^* \backslash \ (X^* is the dual space of X, x^* is a nonzero linear functional) such that x^*\left(x_0\right) \geq x^*(x) for all x \in S, then :H = \ defines a supporting hyperplane. Conversely, if S is a closed set with nonempty interior such that every point on the boundary has a supporting hyperplane, then S is a convex set. The hyperplane in the theorem may not be uniqu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supporting Hyperplane1
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 datase ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boundary (topology)
In topology and mathematics in general, the boundary of a subset of a topological space is the set of points in the closure of not belonging to the interior of . An element of the boundary of is called a boundary point of . The term boundary operation refers to finding or taking the boundary of a set. Notations used for boundary of a set include \operatorname(S), \operatorname(S), and \partial S. Some authors (for example Willard, in ''General Topology'') use the term frontier instead of boundary in an attempt to avoid confusion with a different definition used in algebraic topology and the theory of manifolds. Despite widespread acceptance of the meaning of the terms boundary and frontier, they have sometimes been used to refer to other sets. For example, ''Metric Spaces'' by E. T. Copson uses the term boundary to refer to Hausdorff's border, which is defined as the intersection of a set with its boundary. Hausdorff also introduced the term residue, which is defi ... [...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] |
Supporting Line
In geometry, a supporting line ''L'' of a curve ''C'' in the plane is a line that contains a point of ''C'', but does not separate any two points of ''C''."The geometry of geodesics", Herbert Busemannp. 158/ref> In other words, ''C'' lies completely in one of the two closed half-planes defined by ''L'' and has at least one point on ''L''. Properties There can be many supporting lines for a curve at a given point. When a tangent exists at a given point, then it is the unique supporting line at this point, if it does not separate the curve. Generalizations The notion of supporting line is also discussed for planar shapes. In this case a supporting line may be defined as a line which has common points with the boundary of the shape, but not with its interior."Encyclopedia of Distances", by Michel M. Deza, Elena Dezap. 179/ref> The notion of a supporting line to a planar curve or convex shape can be generalized to n dimension as a supporting hyperplane. Critical support lines If tw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Support Function
In mathematics, the support function ''h''''A'' of a non-empty closed convex set ''A'' in \mathbb^n describes the (signed) distances of supporting hyperplanes of ''A'' from the origin. The support function is a convex function on \mathbb^n. Any non-empty closed convex set ''A'' is uniquely determined by ''h''''A''. Furthermore, the support function, as a function of the set ''A'', is compatible with many natural geometric operations, like scaling, translation, rotation and Minkowski addition. Due to these properties, the support function is one of the most central basic concepts in convex geometry. Definition The support function h_A\colon\mathbb^n\to\mathbb of a non-empty closed convex set ''A'' in \mathbb^n is given by : h_A(x)=\sup\, x\in\mathbb^n; see T. Bonnesen, W. Fenchel, '' Theorie der konvexen Körper,'' Julius Springer, Berlin, 1934. English translation: ''Theory of convex bodies,'' BCS Associates, Moscow, ID, 1987. R. J. Gardner, ''Geometric tomography,'' Cam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supporting Hyperplane3
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 datase ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separating Hyperplane Theorem
In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in ''n''-dimensional Euclidean space. There are several rather similar versions. In one version of the theorem, if both these sets are closed and at least one of them is compact, then there is a hyperplane in between them and even two parallel hyperplanes in between them separated by a gap. In another version, if both disjoint convex sets are open, then there is a hyperplane in between them, but not necessarily any gap. An axis which is orthogonal to a separating hyperplane is a separating axis, because the orthogonal projections of the convex bodies onto the axis are disjoint. The hyperplane separation theorem is due to Hermann Minkowski. The Hahn–Banach separation theorem generalizes the result to topological vector spaces. A related result is the supporting hyperplane theorem. In the context of support-vector machines, the ''optimally separating hyperplane'' or ''maximum-margin hype ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John W
John is a common English name and surname: * John (given name) * John (surname) John may also refer to: New Testament Works * Gospel of John, a title often shortened to John * First Epistle of John, often shortened to 1 John * Second Epistle of John, often shortened to 2 John * Third Epistle of John, often shortened to 3 John People * John the Baptist (died c. AD 30), regarded as a prophet and the forerunner of Jesus Christ * John the Apostle (lived c. AD 30), one of the twelve apostles of Jesus * John the Evangelist, assigned author of the Fourth Gospel, once identified with the Apostle * John of Patmos, also known as John the Divine or John the Revelator, the author of the Book of Revelation, once identified with the Apostle * John the Presbyter, a figure either identified with or distinguished from the Apostle, the Evangelist and John of Patmos Other people with the given name Religious figures * John, father of Andrew the Apostle and Saint Peter * Pope Jo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interior (topology)
In mathematics, specifically in general topology, topology, the interior of a subset of a topological space is the Union (set theory), union of all subsets of that are Open set, open in . A point that is in the interior of is an interior point of . The interior of is the Absolute complement, complement of the closure (topology), closure of the complement of . In this sense interior and closure are Duality_(mathematics)#Duality_in_logic_and_set_theory, dual notions. The exterior of a set is the complement of the closure of ; it consists of the points that are in neither the set nor its boundary (topology), boundary. The interior, boundary, and exterior of a subset together partition of a set, partition the whole space into three blocks (or fewer when one or more of these is empty set, empty). Definitions Interior point If is a subset of a Euclidean space, then is an interior point of if there exists an open ball centered at which is completely contained in . (This is i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the . When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the ''continuous dual space''. Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in tensor analysis with finite-dimensional vector spaces. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis. Early terms for ''dual'' include ''polarer Raum'' ahn 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Vector Space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is also a topological space with the property that the vector space operations (vector addition and scalar multiplication) are also Continuous function, continuous functions. Such a topology is called a and every topological vector space has a Uniform space, uniform topological structure, allowing a notion of uniform convergence and Complete topological vector space, completeness. Some authors also require that the space is a Hausdorff space (although this article does not). One of the most widely studied categories of TVSs are locally convex topological vector spaces. This article focuses on TVSs that are not necessarily locally convex. Banach spaces, Hilbert spaces and Sobolev spaces are other well-known examples of TVSs. Many topological vec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
