# Category: Metric geometry

Hamming distance
In information theory, the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different. In other words, it measures the minimum num
A geodesic metric space is called tree-graded space, with respect to a collection of connected proper subsets called pieces, if any two distinct pieces intersect by at most one point, and every non-tr
Meyer set
In mathematics, a Meyer set or almost lattice is a set relatively dense X of points in the Euclidean plane or a higher-dimensional Euclidean space such that its Minkowski difference with itself is uni
Ultrametric space
In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to . Sometimes the associated metric is also called a non-Archimedean metric or super-metric. Al
Banach–Mazur compactum
In the mathematical study of functional analysis, the Banach–Mazur distance is a way to define a distance on the set of -dimensional normed spaces. With this distance, the set of isometry classes of -
Pseudometric space
In mathematics, a pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. Pseudometric spaces were introduced by Đuro Kurepa in 1934. In
In mathematics, the Cartan–Hadamard theorem is a statement in Riemannian geometry concerning the structure of complete Riemannian manifolds of non-positive sectional curvature. The theorem states that
Quasisymmetric map
In mathematics, a quasisymmetric homeomorphism between metric spaces is a map that generalizes bi-Lipschitz maps. While bi-Lipschitz maps shrink or expand the diameter of a set by no more than a multi
Stretch factor
The stretch factor (i.e., bilipschitz constant) of an embedding measures the factor by which the embedding distorts distances. Suppose that one metric space S is embedded into another metric space T b
Chebyshev distance
In mathematics, Chebyshev distance (or Tchebychev distance), maximum metric, or L∞ metric is a metric defined on a vector space where the distance between two vectors is the greatest of their differen
Kuratowski embedding
In mathematics, the Kuratowski embedding allows one to view any metric space as a subset of some Banach space. It is named after Kazimierz Kuratowski. The statement obviously holds for the empty space
Idempotent measure
In mathematics, an idempotent measure on a is a probability measure that equals its convolution with itself; in other words, an idempotent measure is an idempotent element in the topological semigroup
Gromov product
In mathematics, the Gromov product is a concept in the theory of metric spaces named after the mathematician Mikhail Gromov. The Gromov product can also be used to define δ-hyperbolic metric spaces in
Alexandrov space
In geometry, Alexandrov spaces with curvature ≥ k form a generalization of Riemannian manifolds with sectional curvature ≥ k, where k is some real number. By definition, these spaces are locally compa
In mathematics, the Assouad–Nagata dimension (sometimes simply Nagata dimension) is a notion of dimension for metric spaces, introduced by Jun-iti Nagata in 1958 and reformulated by Patrice Assouad in
Ultralimit
In mathematics, an ultralimit is a geometric construction that assigns to a sequence of metric spaces Xn a limiting metric space. The notion of an ultralimit captures the limiting behavior of finite c
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 fr
Distance (graph theory)
In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path (also called a graph geodesic) connecting them. This is also known as
Equivalence of metrics
In mathematics, two metrics on the same underlying set are said to be equivalent if the resulting metric spaces share certain properties. Equivalence is a weaker notion than isometry; equivalent metri
Great-circle distance
The great-circle distance, orthodromic distance, or spherical distance is the distance along a great circle. It is the shortest distance between two points on the surface of a sphere, measured along t
Minkowski distance
The Minkowski distance or Minkowski metric is a metric in a normed vector space which can be considered as a generalization of both the Euclidean distance and the Manhattan distance. It is named after
Distance geometry
Distance geometry is the branch of mathematics concerned with characterizing and studying sets of points based only on given values of the distances between pairs of points. More abstractly, it is the
Carathéodory metric
In mathematics, the Carathéodory metric is a metric defined on the open unit ball of a complex Banach space that has many similar properties to the Poincaré metric of hyperbolic geometry. It is named
Jung's theorem
In geometry, Jung's theorem is an inequality between the diameter of a set of points in any Euclidean space and the radius of the minimum enclosing ball of that set. It is named after Heinrich Jung, w
Gray isometry
No description available.
Fréchet surface
In mathematics, a Fréchet surface is an equivalence class of parametrized surfaces in a metric space. In other words, a Fréchet surface is a way of thinking about surfaces independently of how they ar
Dilation (metric space)
In mathematics, a dilation is a function from a metric space into itself that satisfies the identity for all points , where is the distance from to and is some positive real number. In Euclidean space
Uniformly disconnected space
In mathematics, a uniformly disconnected space is a metric space for which there exists such that no pair of distinct points can be connected by a -chain.A -chain between and is a sequence of points i
Packing dimension
In mathematics, the packing dimension is one of a number of concepts that can be used to define the dimension of a subset of a metric space. Packing dimension is in some sense dual to Hausdorff dimens
Macbeath region
In mathematics, a Macbeath region is an explicitly defined region in convex analysis on a bounded convex subset of d-dimensional Euclidean space . The idea was introduced by Alexander Macbeath and dub
Urysohn universal space
The Urysohn universal space is a certain metric space that contains all separable metric spaces in a particularly nice manner. This mathematics concept is due to Pavel Urysohn.
Levi–Lechicki theorem
No description available.
Coarse structure
In the mathematical fields of geometry and topology, a coarse structure on a set X is a collection of subsets of the cartesian product X × X with certain properties which allow the large-scale structu
Falconer's conjecture
In geometric measure theory, Falconer's conjecture, named after Kenneth Falconer, is an unsolved problem concerning the sets of Euclidean distances between points in compact -dimensional spaces. Intui
Cayley–Klein metric
In mathematics, a Cayley–Klein metric is a metric on the complement of a fixed quadric in a projective space which is defined using a cross-ratio. The construction originated with Arthur Cayley's essa
Danzer set
In geometry, a Danzer set is a set of points that touches every convex body of unit volume. Ludwig Danzer asked whether it is possible for such a set to have bounded density. Several variations of thi
Growth rate (group theory)
In the mathematical subject of geometric group theory, the growth rate of a group with respect to a symmetric generating set describes how fast a group grows. Every element in the group can be written
Non-positive curvature
In mathematics, spaces of non-positive curvature occur in many contexts and form a generalization of hyperbolic geometry. In the category of Riemannian manifolds, one can consider the sectional curvat
Space of directions
In metric geometry, the space of directions at a point describes the directions of curves that start at the point. It generalizes the tangent space in a differentiable manifold.
Porous set
In mathematics, a porous set is a concept in the study of metric spaces. Like the concepts of meagre and measure zero sets, a porous set can be considered "sparse" or "lacking bulk"; however, porous s
Frostman lemma
In mathematics, and more specifically, in the theory of fractal dimensions, Frostman's lemma provides a convenient tool for estimating the Hausdorff dimension of sets. Lemma: Let A be a Borel subset o
Effective dimension
In mathematics, effective dimension is a modification of Hausdorff dimension and other fractal dimensions that places it in a computability theory setting. There are several variations (various notion
Covering number
In mathematics, a covering number is the number of spherical balls of a given size needed to completely cover a given space, with possible overlaps. Two related concepts are the packing number, the nu
Taxicab geometry
A taxicab geometry or a Manhattan geometry is a geometry in which the usual distance function or metric of Euclidean geometry is replaced by a new metric in which the distance between two points is th
Convex cap
A convex cap, also known as a convex floating body or just floating body, is a well defined structure in mathematics commonly used in convex analysis for approximating convex shapes. In general it can
Ball (mathematics)
In mathematics, a ball is the solid figure bounded by a sphere; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (ex
In mathematics — specifically, in fractal geometry — the Assouad dimension is a definition of fractal dimension for subsets of a metric space. It was introduced by in his 1977 PhD thesis and later pub
Distance matrix
In mathematics, computer science and especially graph theory, a distance matrix is a square matrix (two-dimensional array) containing the distances, taken pairwise, between the elements of a set. Depe
Chow–Rashevskii theorem
In sub-Riemannian geometry, the Chow–Rashevskii theorem (also known as Chow's theorem) asserts that any two points of a connected sub-Riemannian manifold, endowed with a bracket generating distributio
T-theory
T-theory is a branch of discrete mathematics dealing with analysis of trees and discrete metric spaces.
Wasserstein metric
In mathematics, the Wasserstein distance or Kantorovich–Rubinstein metric is a distance function defined between probability distributions on a given metric space . It is named after Leonid Vaseršteĭn
Intrinsic metric
In the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are at a given distance from each other, it is natural to expect that one should be able
Weyl distance function
In combinatorial geometry, the Weyl distance function is a function that behaves in some ways like the distance function of a metric space, but instead of taking values in the positive real numbers, i
Distance set
In geometry, the distance set of a collection of points is the set of distances between distinct pairs of points. Thus, it can be seen as the generalization of a difference set, the set of distances (
Word metric
In group theory, a word metric on a discrete group is a way to measure distance between any two elements of . As the name suggests, the word metric is a metric on , assigning to any two elements , of
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-d
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
Polyhedral space
Polyhedral space is a certain metric space. A (Euclidean) polyhedral space is a (usually finite) simplicial complex in which every simplex has a flat metric. (Other spaces of interest are spherical an
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 tigh
Kirszbraun theorem
In mathematics, specifically real analysis and functional analysis, the Kirszbraun theorem states that if U is a subset of some Hilbert space H1, and H2 is another Hilbert space, and is a Lipschitz-co
Equilateral dimension
In mathematics, the equilateral dimension of a metric space is the maximum size of any subset of the space whose points are all at equal distances to each other. Equilateral dimension has also been ca
Gilbert–Pollack conjecture
No description available.
Tits metric
In mathematics, the Tits metric is a metric defined on the ideal boundary of an Hadamard space (also called a complete CAT(0) space). It is named after Jacques Tits.
Convex metric space
In mathematics, convex metric spaces are, intuitively, metric spaces with the property any "segment" joining two points in that space has other points in it besides the endpoints. Formally, consider a
Bellman's lost in a forest problem
Bellman's lost-in-a-forest problem is an unsolved minimization problem in geometry, originating in 1955 by the American applied mathematician Richard E. Bellman. The problem is often stated as follows
Cauchy sequence
In mathematics, a Cauchy sequence (French pronunciation: ​[koʃi]; English: /ˈkoʊʃiː/ KOH-shee), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other a
Doubling space
In mathematics, a metric space X with metric d is said to be doubling if there is some doubling constant M > 0 such that for any x ∈ X and r > 0, it is possible to cover the ball B(x, r) = {y | d(x, y
Isometry group
In mathematics, the isometry group of a metric space is the set of all bijective isometries (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the function composition
Geodesic bicombing
In metric geometry, a geodesic bicombing distinguishes a class of geodesics of a metric space. The study of metric spaces with distinguished geodesics traces back to the work of the mathematician Herb
Curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a
Asymptotic dimension
In metric geometry, asymptotic dimension of a metric space is a large-scale analog of Lebesgue covering dimension. The notion of asymptotic dimension was introduced by Mikhail Gromov in his 1993 monog
Canberra distance
The Canberra distance is a numerical measure of the distance between pairs of points in a vector space, introduced in 1966and refined in 1967 by Godfrey N. Lance and William T. Williams. It is a weigh
Metric derivative
In mathematics, the metric derivative is a notion of derivative appropriate to parametrized paths in metric spaces. It generalizes the notion of "speed" or "absolute velocity" to spaces which have a n
Metric map
In the mathematical theory of metric spaces, a metric map is a function between metric spaces that does not increase any distance (such functions are always continuous).These maps are the morphisms in
Švarc–Milnor lemma
In the mathematical subject of geometric group theory, the Švarc–Milnor lemma (sometimes also called Milnor–Švarc lemma, with both variants also sometimes spelling Švarc as Schwarz) is a statement whi
Probabilistic metric space
In mathematics, probabilistic metric spaces are a generalization of metric spaces where the distance no longer takes values in the non-negative real numbers R ≥ 0, but in distribution functions. Let D
Totally bounded space
In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered b
Generalised metric
In mathematics, the concept of a generalised metric is a generalisation of that of a metric, in which the distance is not a real number but taken from an arbitrary ordered field. In general, when we d
GNRS conjecture
In theoretical computer science and metric geometry, the GNRS conjecture connects the theory of graph minors, the stretch factor of embeddings, and the approximation ratio of multi-commodity flow prob
Positively separated sets
In mathematics, two non-empty subsets A and B of a given metric space (X, d) are said to be positively separated if the infimum (Some authors also specify that A and B should be disjoint sets; however
Fréchet distance
In mathematics, the Fréchet distance is a measure of similarity between curves that takes into account the location and ordering of the points along the curves. It is named after Maurice Fréchet.
Gromov–Hausdorff convergence
In mathematics, Gromov–Hausdorff convergence, named after Mikhail Gromov and Felix Hausdorff, is a notion for convergence of metric spaces which is a generalization of Hausdorff convergence.
Johnson–Lindenstrauss lemma
In mathematics, the Johnson–Lindenstrauss lemma is a result named after William B. Johnson and Joram Lindenstrauss concerning low-distortion embeddings of points from high-dimensional into low-dimensi
Hausdorff dimension
In mathematics, Hausdorff dimension is a measure of roughness, or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff
Delone set
In the mathematical theory of metric spaces, ε-nets, ε-packings, ε-coverings, uniformly discrete sets, relatively dense sets, and Delone sets (named after Boris Delone) are several closely related def
Lévy–Prokhorov metric
In mathematics, the Lévy–Prokhorov metric (sometimes known just as the Prokhorov metric) is a metric (i.e., a definition of distance) on the collection of probability measures on a given metric space.
Metric outer measure
In mathematics, a metric outer measure is an outer measure μ defined on the subsets of a given metric space (X, d) such that for every pair of positively separated subsets A and B of X.
Complete metric space
In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if there are n
Complex geodesic
In mathematics, a complex geodesic is a generalization of the notion of geodesic to complex spaces.
Quasi-isometry
In mathematics, a quasi-isometry is a function between two metric spaces that respects large-scale geometry of these spaces and ignores their small-scale details. Two metric spaces are quasi-isometric
Elliptic geometry
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. How
Caristi fixed-point theorem
In mathematics, the Caristi fixed-point theorem (also known as the Caristi–Kirk fixed-point theorem) generalizes the Banach fixed-point theorem for maps of a complete metric space into itself. Caristi
Reshetnyak gluing theorem
In metric geometry, the Reshetnyak gluing theorem gives information on the structure of a geometric object built by using as building blocks other geometric objects, belonging to a well defined class.
Banach fixed-point theorem
In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem) is an important tool in the theory of metric spaces; it guarantees the exi
Hausdorff measure
In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of out
Aleksandrov–Rassias problem
The theory of isometries in the framework of Banach spaces has its beginning in a paper by Stanisław Mazur and Stanisław M. Ulam in 1932. They proved the Mazur–Ulam theorem stating that every isometry
Gromov's theorem on groups of polynomial growth
In geometric group theory, Gromov's theorem on groups of polynomial growth, first proved by Mikhail Gromov, characterizes finitely generated groups of polynomial growth, as those groups which have nil
Kuratowski convergence
In mathematics, Kuratowski convergence is a notion of convergence for sequences (or, more generally, nets) of compact subsets of metric spaces, named after Kazimierz Kuratowski. Intuitively, the Kurat
Laplace functional
In probability theory, a Laplace functional refers to one of two possible mathematical functions of functions or, more precisely, functionals that serve as mathematical tools for studying either point
Hilbert metric
In mathematics, the Hilbert metric, also known as the Hilbert projective metric, is an explicitly defined distance function on a bounded convex subset of the n-dimensional Euclidean space Rn. It was i
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,
Hutchinson metric
In mathematics, the Hutchinson metric otherwise known as Kantorovich metric is a function which measures "the discrepancy between two images for use in fractal image processing" and "can also be appli
Dimension function
In mathematics, the notion of an (exact) dimension function (also known as a gauge function) is a tool in the study of fractals and other subsets of metric spaces. Dimension functions are a generalisa
Hopf–Rinow theorem
Hopf–Rinow theorem is a set of statements about the geodesic completeness of Riemannian manifolds. It is named after Heinz Hopf and his student Willi Rinow, who published it in 1931. Stefan Cohn-Vosse
Hyperbolic group
In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a word hyperbolic group or Gromov hyperbolic group, is a finitely generated group equipped with a word metr
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
Conformal dimension
In mathematics, the conformal dimension of a metric space X is the infimum of the Hausdorff dimension over the of X, that is, the class of all metric spaces quasisymmetric to X.
Product metric
In mathematics, a product metric is a metric on the Cartesian product of finitely many metric spaces which metrizes the product topology. The most prominent product metrics are the p product metrics f
Random polytope
In mathematics, a random polytope is a structure commonly used in convex analysis and the analysis of linear programs in d-dimensional Euclidean space . Depending on use the construction and definitio
Intrinsic flat distance
In mathematics, the intrinsic flat distance is a notion for distance between two Riemannian manifolds which is a generalization of Federer and Fleming's flat distance between submanifolds and integral
Wijsman convergence
Wijsman convergence is a variation of Hausdorff convergence suitable for work with unbounded sets.Intuitively, Wijsman convergence is to convergence in the Hausdorff metric as pointwise convergence is
Category of metric spaces
In category theory, Met is a category that has metric spaces as its objects and metric maps (continuous functions between metric spaces that do not increase any pairwise distance) as its morphisms. Th
Lévy metric
In mathematics, the Lévy metric is a metric on the space of cumulative distribution functions of one-dimensional random variables. It is a special case of the Lévy–Prokhorov metric, and is named after
Hausdorff distance
In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty c
Classical Wiener space
In mathematics, classical Wiener space is the collection of all continuous functions on a given domain (usually a subinterval of the real line), taking values in a metric space (usually n-dimensional
Discrete metric
No description available.
Motion (geometry)
In geometry, a motion is an isometry of a metric space. For instance, a plane equipped with the Euclidean distance metric is a metric space in which a mapping associating congruent figures is a motion
CAT(k) space
In mathematics, a space, where is a real number, is a specific type of metric space. Intuitively, triangles in a space are "slimmer" than corresponding "model triangles" in a standard space of constan
Glossary of Riemannian and metric geometry
This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology. The following articles may also be useful; they either con
Flat convergence
In mathematics, flat convergence is a notion for convergence of submanifolds of Euclidean space. It was first introduced by Hassler Whitney in 1957, and then extended to integral currents by Federer a
Sub-Riemannian manifold
In mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold. Roughly speaking, to measure distances in a sub-Riemannian manifold, you are allowed to go only
Heine–Cantor theorem
In mathematics, the Heine–Cantor theorem, named after Eduard Heine and Georg Cantor, states that if is a continuous function between two metric spaces and , and is compact, then is uniformly continuou
Beer's theorem
No description available.
Contraction mapping
In mathematics, a contraction mapping, or contraction or contractor, on a metric space (M, d) is a function f from M to itself, with the property that there is some real number such that for all x and
Comparison triangle
Define as the 2-dimensional metric space of constant curvature . So, for example, is the Euclidean plane, is the surface of the unit sphere, and is the hyperbolic plane. Let be a metric space. Let be
Euclidean distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points.It can be calculated from the Cartesian coordinates of the points us
Real-valued function
In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a
Riemannian circle
In metric space theory and Riemannian geometry, the Riemannian circle is a great circle with a characteristic length. It is the circle equipped with the intrinsic Riemannian metric of a compact one-di
Beckman–Quarles theorem
In geometry, the Beckman–Quarles theorem, named after Frank S. Beckman and Donald A. Quarles Jr., states that if a transformation of the Euclidean plane or a higher-dimensional Euclidean space preserv
Laakso space
In mathematical analysis and metric geometry, Laakso spaces are a class of metric spaces which are fractal, in the sense that they have non-integer Hausdorff dimension, but that admit a notion of diff