Intrinsic metric

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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 one point to another 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 length of all paths from one point to the other. A metric space is a length metric space if the intrinsic metric agrees with the original metric of the space.


Let be a metric space. 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 . 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





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