# 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 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.

## Definitions

Let ${\displaystyle (M,d)}$ be a metric space. 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 ${\displaystyle y}$.

${\displaystyle \gamma \colon [0,1]\rightarrow M}$

with ${\displaystyle \gamma (0)=x}$ and ${\displaystyle \gamma (1)=y}$. The length of such a path is defined as explained for rectifiable curves. We set ${\displaystyle d_{\text{I}}(x,y)=\infty }$ if there is no path of finite length from ${\displaystyle x}$ to ${\displaystyle y}$. If

${\displaystyle d_{\text{I}}(x,y)=d(x,y)}$

for all points ${\displaystyle x}$ and ${\displaystyle y}$ in ${\displaystyle M}$, we say that ${\displaystyle (M,d)}$ is a length space or a path metric space and the metric ${\displaystyle d}$ is intrinsic.

We say that the metric ${\displaystyle d}$ has approximate midpoints if for any ${\displaystyle \varepsilon >0}$ and any pair of points ${\displaystyle x}$ and ${\displaystyle y}$ in ${\displaystyle M}$ there exists ${\displaystyle c}$ in ${\displaystyle M}$ such that ${\displaystyle d(x,c)}$ and ${\displaystyle d(c,y)}$ are both smaller than

${\displaystyle {d(x,y)}/{2}+\varepsilon }$.

## References

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