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.

Let ${\displaystyle (M,d)}$ be a metric space, i.e., ${\displaystyle M}$ is a collection of points (such as all of the points in the plane, or all points on the circle) and ${\displaystyle d(x,y)}$ is a function that provides us with the distance between points ${\displaystyle x,y\in M}$. We define a new metric ${\displaystyle d_{\text{I}}}$ on ${\displaystyle M}$, known as the induced intrinsic metric, as follows: ${\displaystyle d_{\text{I}}(x,y)}$ is the infimum of the lengths of all paths from ${\displaystyle x}$ to ${\displaystyle y}$.

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