Metric geometry

right|thumb|A parabola, one of the simplest curves, after (straight) lines

in mathematics, space bounded by a sphere

sequence whose elements become arbitrarily close to each other

inequality of the form |x + y| ≤ |x| + |y|

conventional distance in mathematics and physics

thumb|upright=1.4|A Function composition|composition of two opposite isometries is a direct isometry. A reflection in a line is an opposite isometry, like (reflection w.r.t the center diagonal line) or (reflection w.r.t the right diagonal line) on the image. Translation is a direct isometry: a rigid motion.

number of bits that differ between two strings

metric space in which cauchy sequence converges to an element of the space

non-Euclidean geometry

invariant

theorem about metric spaces

shortest distance between two points along the surface of a sphere

type of metric geometry

function whose range is a subset of the real numbers

distance between vectors defined by the maximum difference between coordinates

theorem

in graph theory, the minimum number of edges in a path connecting two vertices

function reducing distance between all points

a type of metric space in which the triangle inequality is replaced by a stronger inequality using max in place of addition

generalization of a metric space in which the distance between two distinct points can be zero

mathematical distance between two subsets of a metric space

fractal measurement

distance between vectors or points computed as the pth root of the sum of pth powers of coordinate differences

theorem relating the diameter of a point set to the minimum radius of an enclosing ball

square matrix (two-dimensional array) containing the distances, taken pairwise, between the elements of a set. Depending upon the application involved, the distance being used to define this matrix may or may not be a metric

uniform space which has finite covers consisting of "arbitrarily small" sets

theorem that, for a Riemannian manifold, geodesic completeness is equivalent to completeness as a metric space

automorphism group of a metric space

Mathematical result

function between metric spaces that does not increase any distance

distance function defined between probability distributions

Mathematical concept

On the structure of complete Riemannian manifolds of non-positive sectional curvature

certain metric on space of finite measures

Concept in geometry/topology

mathematical theorem related to real and functional analysis

isometry of a metric space

a notion for convergence of metric spaces

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 if there exists a quasi-isometry between them. The property of being quasi-isometric behaves like an equivalence relation on the class of metric spaces.

theorem

lemma in geometric group theory, giving sufficient conditions for when a group equipped with an isometric action on a metric space is quasi-isometric to the metric space

well-spaced set of points in a metric space

concept from the theory of metric spaces

way to measure distance between any two elements of group (in group theory)

mathematical category with metric spaces as its objects and distance-non-increasing maps as its morphisms

type of generalization of a Riemannian manifold

Concept in mathematics

function from a metric space into itself

family of sets in geometry and topology to measure large-scale properties of a space

theorem in geometric group theory

metric space with double measurement