The least distorted curves and surfaces

Every subset {A\subset \mathbb R^n} inherits the metric from {\mathbb R^n}, namely {d(a,b)=|a-b|}. But we can also consider the intrinsic metric on {A}, defined as follows: {\rho_A(a,b)} is the infimum of the lengths of curves that connect {a} to {b} within {A}. Let’s assume there is always such a curve of finite length, and therefore {\rho_A} is always finite. All the properties of a metric hold, and we also have {|a-b|\le \rho_A(a,b)} for all {a,b\in A}.

If {A} happens to be convex, then {\rho_A(a,b)=|a-b|} because any two points are joined by a line segment. There are also some nonconvex sets for which {\rho_A} coincides with the Euclidean distance: for example, the punctured plane {\mathbb R^2\setminus \{(0,0)\}}. Although we can’t always get from {a} to {b} in a straight line, the required detour can be as short as we wish.

On the other hand, for the set {A=\{(x,y)\in \mathbb R^2 : y\le |x|\}} the intrinsic distance is sometimes strictly greater than Euclidean distance.

Nontrivial distortion
Oops, the equation was supposed to be y=|x|, without the square

For example, the shortest curve from {(-1,1)} to {(1,1)} has length {2\sqrt{2}}, while the Euclidean distance is {2}. 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 {A} as the supremum of the ratios {\rho_A(a,b)/|a-b|} over all pairs of distinct points {a,b\in A}. (Another term in use for this concept: optimal constant of quasiconvexity.) So, the distortion of the set {\{(x,y) : y\le |x|\}} is {\sqrt{2}}.

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 {\pi/2}. That is, the least distorted closed curve is the circle, for which the half-length/diameter ratio is exactly {\pi/2}.

Distortion of a closed curve
Distortion of a closed curve

Here is the proof. Parametrize the curve by arclength: {\gamma\colon [0,L]\rightarrow \mathbb R^n}. For {0\le t\le L/2} define {\Gamma(t)=\gamma(t )-\gamma(t+L/2) } and let {r=\min_t|\Gamma(t)|}. The curve {\Gamma} connects two antipodal points of magnitude at least {r}, and stays outside of the open ball of radius {r} centered at the origin. Therefore, its length is at least {\pi r} (projection onto a convex subset does not increase the length). On the other hand, {\Gamma} is a 2-Lipschitz map, which implies {\pi r\le 2(L/2)}. Thus, {r\le L/\pi}. Take any {t} that realizes the minimum of {|\Gamma|}. The points {a=\gamma(t)} and {b=\gamma(t+L/2)} satisfy {|a-b|\le L/\pi} and {\rho_A(a,b)=L/2}. Done.

Follow-up question: what are the least distorted closed surfaces (say, in {\mathbb R^3})? It’s natural to expect that a sphere, with distortion {\pi/2}, 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 {\mathbb R^3} with distortion less than { \pi/2}. (Here, “convex” means the surface bounds a convex solid.)

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