Every subset inherits the metric from , namely . But we can also consider the intrinsic metric on , defined as follows: is the infimum of the lengths of curves that connect to within . Let’s assume there is always such a curve of finite length, and therefore is always finite. All the properties of a metric hold, and we also have for all .
If happens to be convex, then because any two points are joined by a line segment. There are also some nonconvex sets for which coincides with the Euclidean distance: for example, the punctured plane . Although we can’t always get from to in a straight line, the required detour can be as short as we wish.
On the other hand, for the set the intrinsic distance is sometimes strictly greater than Euclidean distance.
For example, the shortest curve from to has length , while the Euclidean distance is . This is the worst ratio for pairs of points in this set, although proving this claim would be a bit tedious. Following Gromov (Metric structures on Riemannian and non-Riemannian spaces), define the distortion of as the supremum of the ratios over all pairs of distinct points . (Another term in use for this concept: optimal constant of quasiconvexity.) So, the distortion of the set is .
Gromov observed (along with posing the Knot Distortion Problem) that every simple closed curve in a Euclidean space (of any dimension) has distortion at least . That is, the least distorted closed curve is the circle, for which the half-length/diameter ratio is exactly .
Here is the proof. Parametrize the curve by arclength: . For define and let . The curve connects two antipodal points of magnitude at least , and stays outside of the open ball of radius centered at the origin. Therefore, its length is at least (projection onto a convex subset does not increase the length). On the other hand, is a 2-Lipschitz map, which implies . Thus, . Take any that realizes the minimum of . The points and satisfy and . Done.
Follow-up question: what are the least distorted closed surfaces (say, in )? It’s natural to expect that a sphere, with distortion , is the least distorted. But this is false. An exercise from Gromov’s book (which I won’t spoil): Find a closed convex surface in with distortion less than . (Here, “convex” means the surface bounds a convex solid.)