Intrinsic Metric

(Metric Geometry)

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sciences
Metric Geometry
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What is 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 to get from the first point to the second along a path whose arclength is equal to (or very close to) that distance. The distance between two points of a metric space relative to the intrinsic metric is defined as the infimum of the lengths of all paths from the first point to the second. A metric space is a length metric space if the intrinsic metric agrees with the original metric of the space. If the space has the stronger property that there always exists a path that achieves the infimum of length (a geodesic) then it may be called a geodesic metric space or geodesic space. For instance, the Euclidean plane is a geodesic space, with line segments as its geodesics. The Euclidean plane with the origin removed is not geodesic, but is still a length metric space.

Technology Types

metric geometry

Synonyms

Length metric spaceLength spaceLength-metric spaceMetric convexity

Translations

Espace de longueur (fr)Innere Metrik (de)Metrica intrinseca (it)Внутренняя метрика (ru)Внутрішня метрика (uk)길이 거리 공간 (ko)

Tech Info


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 — Date merged: 11/6/2021, 1:32:52 PM
 — Date scraped: 5/20/2021, 6:07:17 PM