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. 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 ( 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 ( x , y ) {displaystyle d(x,y)} is a function that provides us with the distance between points x , y ∈ M {displaystyle x,yin M} . We define a new metric d I {displaystyle d_{ ext{I}}} on M {displaystyle M} , known as the induced intrinsic metric, as follows: d I ( x , y ) {displaystyle d_{ ext{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 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 explained for rectifiable curves. We set d I ( x , y ) = ∞ {displaystyle d_{ ext{I}}(x,y)=infty } if there is no path of finite length from x {displaystyle x} to y {displaystyle y} . If for all points x {displaystyle x} and y {displaystyle y} in M {displaystyle M} , we say that ( M , d ) {displaystyle (M,d)} is a length space or a path metric space and the metric d {displaystyle d} is intrinsic. We say that the metric d {displaystyle d} has approximate midpoints if for any ε > 0 {displaystyle varepsilon >0} and any pair of points x {displaystyle x} and y {displaystyle y} in M {displaystyle M} there exists c {displaystyle c} in M {displaystyle M} such that d ( x , c ) {displaystyle d(x,c)} and d ( c , y ) {displaystyle d(c,y)} are both smaller than