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.

Definitions

Let (M, d) be a metric space, i.e., M is a collection of points (such as all of the points in the plane, or all points on the circle) and d(x,y) is a function that provides us with the distance between points x,y\in M. We define a new metric d_\text{I} on M, known as the induced intrinsic metric, as follows: is the infimum of the lengths of all paths from x to y. Here, a path from x to y 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 x to y (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 x and y in M, we say that (M, d) is a length space or a path metric space and the metric d is intrinsic. We say that the metric d has approximate midpoints if for any and any pair of points x and y in M there exists c in M such that d(x,c) and d(c,y) are both smaller than

Examples

Properties