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Tight Span
In metric geometry, the metric envelope or tight span of a metric space ''M'' is an injective metric space into which ''M'' can be embedded. In some sense it consists of all points "between" the points of ''M'', analogous to the convex hull of a point set in a Euclidean space. The tight span is also sometimes known as the injective envelope or hyperconvex hull of ''M''. It has also been called the injective hull, but should not be confused with the injective hull of a module in algebra, a concept with a similar description relative to the category of ''R''-modules rather than metric spaces. The tight span was first described by , and it was studied and applied by Holsztyński in the 1960s. It was later independently rediscovered by and ; see for this history. The tight span is one of the central constructions of T-theory. Definition The tight span of a metric space can be defined as follows. Let (''X'',''d'') be a metric space, and let ''T''(''X'') be the set of extremal functio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Geometry
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance and t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Metric , a random variable that can be counted
*Discrete space, a simple example of a topological space
*Discrete spline interpolation, the discrete analog of ordinary spline interpolation
*Discrete time, ...
Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a group with the discrete topology *Discrete category, category whose only arrows are identity arrows *Discrete mathematics, the study of structures without continuity *Discrete optimization, a branch of optimization in applied mathematics and computer science *Discrete probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Space Aimed At Its Subspace
In mathematics, a metric space aimed at its subspace is a categorical construction that has a direct geometric meaning. It is also a useful step toward the construction of the ''metric envelope'', or tight span, which are basic (injective) objects of the category of metric spaces. Following , a notion of a metric space ''Y'' aimed at its subspace ''X'' is defined. Informal introduction Informally, imagine terrain ''Y'', and its part ''X'', such that wherever in ''Y'' you place a sharpshooter, and an apple at another place in ''Y'', and then let the sharpshooter fire, the bullet will go through the apple and will always hit a point of ''X'', or at least it will fly arbitrarily close to points of ''X'' – then we say that ''Y'' is aimed at ''X''. A priori, it may seem plausible that for a given ''X'' the superspaces ''Y'' that aim at ''X'' can be arbitrarily large or at least huge. We will see that this is not the case. Among the spaces which aim at a subspace isometric to ''X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedral Complex
In mathematics, a polyhedral complex is a set of polyhedra in a real vector space that fit together in a specific way. Polyhedral complexes generalize simplicial complexes and arise in various areas of polyhedral geometry, such as tropical geometry, splines and hyperplane arrangements. Definition A polyhedral complex \mathcal is a set of polyhedra that satisfies the following conditions: :1. Every face of a polyhedron from \mathcal is also in \mathcal. :2. The intersection of any two polyhedra \sigma_1, \sigma_2 \in \mathcal is a face of both \sigma_1 and \sigma_2. Note that the empty set is a face of every polyhedron, and so the intersection of two polyhedra in \mathcal may be empty. Examples * Tropical varieties are polyhedral complexes satisfying a certain ''balancing condition''. * Simplicial complexes are polyhedral complexes in which every polyhedron is a simplex. * Voronoi diagrams. * Splines. Fans A fan is a polyhedral complex in which every polyhedron is a cone fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Convex Hull
In geometry, a set is defined to be orthogonally convex if, for every line that is parallel to one of standard basis vectors, the intersection of with is empty, a point, or a single segment. The term "orthogonal" refers to corresponding Cartesian basis and coordinates in Euclidean space, where different basis vectors are perpendicular, as well as corresponding lines. Unlike ordinary convex sets, an orthogonally convex set is not necessarily connected. The orthogonal convex hull of a set is the intersection of all connected orthogonally convex supersets of . These definitions are made by analogy with the classical theory of convexity, in which is convex if, for every line , the intersection of with is empty, a point, or a single segment. Orthogonal convexity restricts the lines for which this property is required to hold, so every convex set is orthogonally convex but not vice versa. For the same reason, the orthogonal convex hull itself is a subset of the convex hul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manhattan Distance
A taxicab geometry or a Manhattan geometry is a geometry whose usual distance function or Metric (mathematics), metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian coordinates. The taxicab metric is also known as rectilinear distance, ''L''1 distance, ''L''1 distance or \ell_1 norm (see Lp space, ''Lp'' space), Snake (video game), snake distance, city block distance, Manhattan distance or Manhattan length. The latter names refer to the rectilinear street layout on the island of Manhattan, where the shortest path a taxi travels between two points is the sum of the absolute values of distances that it travels on avenues and on streets. The geometry has been used in regression analysis since the 18th century, and is often referred to as Lasso (statistics), LASSO. The geometric interpretation dates to non-Euclidean geometry of the 19th century and is due to Hermann Minkowski. In \mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' meaning "equal", and μέτρον ''metron'' meaning "measure". Introduction Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a transformation which maps elements to the same or another metric space such that the distance between the image elements in the new metric space is equal to the distance between the elements in the original metric space. In a two-dimensional or three-dimensional Euclidean space, two geometric figures are congruent if they are related by an isometry; the isometry that relates them is either a rigid motion (translation or rotation), or a composition of a rigid motion and a reflection. Isometries are often used in constructions where one space i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Injective Metric Space
In metric geometry, an injective metric space, or equivalently a hyperconvex metric space, is a metric space with certain properties generalizing those of the real line and of L∞ distances in higher-dimensional vector spaces. These properties can be defined in two seemingly different ways: hyperconvexity involves the intersection properties of closed balls in the space, while injectivity involves the isometric embeddings of the space into larger spaces. However it is a theorem of that these two different types of definitions are equivalent. Hyperconvexity A metric space X is said to be hyperconvex if it is convex and its closed balls have the binary Helly property. That is: #Any two points x and y can be connected by the isometric image of a line segment of length equal to the distance between the points (i.e. X is a path space). #If F is any family of closed balls _r(p) = \ such that each pair of balls in F meets, then there exists a point x common to all the balls in F. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Metric Space
In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations (depending quantitatively on a nonnegative real number δ) between points. The definition, introduced by Mikhael Gromov, generalizes the metric properties of classical hyperbolic geometry and of trees. Hyperbolicity is a large-scale property, and is very useful to the study of certain infinite groups called Gromov-hyperbolic groups. Definitions In this paragraph we give various definitions of a \delta-hyperbolic space. A metric space is said to be (Gromov-) hyperbolic if it is \delta-hyperbolic for some \delta > 0. Definition using the Gromov product Let (X,d) be a metric space. The Gromov product of two points y, z \in X with respect to a third one x \in X is defined by the formula: :(y,z)_x = \frac 1 2 \left( d(x, y) + d(x, z) - d(y, z) \right). Gromov's definition of a hyperbolic metric space is then as follows: X is \delta-hyperbolic if and only if all x,y,z,w \in X satisfy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reverse Triangle Inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality. If , , and are the lengths of the sides of the triangle, with no side being greater than , then the triangle inequality states that :z \leq x + y , with equality only in the degenerate case of a triangle with zero area. In Euclidean geometry and some other geometries, the triangle inequality is a theorem about distances, and it is written using vectors and vector lengths ( norms): :\, \mathbf x + \mathbf y\, \leq \, \mathbf x\, + \, \mathbf y\, , where the length of the third side has been replaced by the vector sum . When and are real numbers, they can be viewed as vectors in , and the trian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' meaning "equal", and μέτρον ''metron'' meaning "measure". Introduction Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a transformation which maps elements to the same or another metric space such that the distance between the image elements in the new metric space is equal to the distance between the elements in the original metric space. In a two-dimensional or three-dimensional Euclidean space, two geometric figures are congruent if they are related by an isometry; the isometry that relates them is either a rigid motion (translation or rotation), or a composition of a rigid motion and a reflection. Isometries are often used in constructions where one space i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
