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. 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 ) {\displaystyle (M,d)} be a metric space, i.e., M {\displaystyle M} is a collection of points (such as all of the points in the plane, or all points on the circle) and d {\displaystyle d} is a function that outputs the distance between points in M {\displaystyle M} . We define a new metric d I {\displaystyle d_{\text{I}}} on M {\displaystyle M} , known as the induced intrinsic metric, as follows:
d I ( x , y ) {\displaystyle d_{\text{I}}(x,y)} is the infimum of the lengths of all paths from x {\displaystyle x} to y {\displaystyle y} . Here, a path from x {\displaystyle x} to y {\displaystyle y} is a continuous map
γ : [ 0 , 1 ] → M {\displaystyle \gamma \colon [0,1]\rightarrow M}
with γ ( 0 ) = x {\displaystyle \gamma (0)=x} and γ ( 1 ) = y {\displaystyle \gamma (1)=y} . The length of such a path is defined as follows: to each finite partition
P = { 0 = x 0 < x 1 < . . . < x n = 1 } {\displaystyle P=\{0=x_{0}<x_{1}<...<x_{n}=1\}}
of the interval [ 0 , 1 ] {\displaystyle [0,1]} , consider the sum
Σ ( P ) = ∑ k = 0 n − 1 d ( γ ( x k ) , γ ( x k + 1 ) ) . {\displaystyle \Sigma (P)=\sum _{k=0}^{n-1}d(\gamma (x_{k}),\gamma (x_{k+1})).}
We then define the length of γ {\displaystyle \gamma } to be
ℓ ( γ ) = sup P ∈ P Σ ( P ) , {\displaystyle \ell (\gamma )=\sup _{P\in {\mathfrak {P}}}\Sigma (P),}
where P {\displaystyle {\mathfrak {P}}} is the set of finite partitions of [ 0 , 1 ] {\displaystyle [0,1]} . If the supremum is finite, we call γ {\displaystyle \gamma } a rectifiable curve. Note that d I ( x , y ) = ∞ {\displaystyle d_{\text{I}}(x,y)=\infty } if there is no path from x {\displaystyle x} to y {\displaystyle y} since the infimum of the empty set within the closed interval [0,+∞] is +∞. The mapping d ↦ d I {\textstyle d\mapsto d_{\text{I}}} is idempotent, i.e.
( d I ) I = d I . {\displaystyle (d_{\text{I}})_{\text{I}}=d_{\text{I}}.}
If
d I ( x , y ) = d ( x , y ) {\displaystyle d_{\text{I}}(x,y)=d(x,y)}
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