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Concept
Intrinsic metric
Summary
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 is 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.
Let be a metric space, i.e., is a collection of points (such as all of the points in the plane, or all points on the circle) and is a function that provides us with the distance between points . We define a new metric on , known as the induced intrinsic metric, as follows:
is the infimum of the lengths of all paths from to .
Here, a path from to is a continuous map
with and . The length of such a path is defined as explained for rectifiable curves. We set if there is no path of finite length from to (this is consistent with the infimum definition since the infimum of the empty set within the closed interval [0,+∞] is +∞).
The mapping is idempotent, i.e.
If
for all points and in , we say that is a length space or a path metric space and the metric is intrinsic.
We say that the metric has approximate midpoints if for any and any pair of points and in there exists in such that and are both smaller than
Euclidean space with the ordinary Euclidean metric is a path metric space. is as well.
The unit circle with the metric inherited from the Euclidean metric of (the chordal metric) is not a path metric space.
Official source
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