Noncontracting Jordan curves

A simple closed curve {\Gamma} on the plane can be parameterized by a homeomorphism {f\colon \mathbb T\to \Gamma} in infinitely many ways. It is natural to look for “nice” parameterizations, like nonexpanding ones in the previous post. This time, let us look for noncontracting parameterizations: {|f(a)-f(b)|\ge |a-b|} for all {a, b\in \mathbb T}. Note that Euclidean distance is used here, not arclength. And we still want {f} to be a homeomorphism. Its noncontracting property simply means that the inverse {f^{-1}} is nonexpanding aka 1-Lipschitz.

What are some necessary conditions for the existence of a noncontracting parameterization? We can mimic the three from the earlier post Nonexpanding Jordan curves, with similar proofs:

  1. The curve must have length at least {2\pi}.
  2. The curve must have diameter at least 2.
  3. The curve must enclose some disk of radius 1. (Apply Kirszbraun’s theorem to {f^{-1}} and note that the resulting Lipschitz map G of the plane will attain 0 somewhere, by a topological degree / winding number argument. Any point where G = 0 works as the center of such a disk.)

This time, the 3rd item supersedes both 1 and 2. Yet, the condition it presents is not sufficient for the existence of a noncontracting parameterization. A counterexample is a disk with a “comb-over”:

combover

Indeed, suppose that {g=f^{-1}} is a nonexpanding map from this curve onto the unit circle. Let {1 = A < B < C} be the three points at which the curve meets the positive x-axis. Since every point of the curve is within small distance from its arc AB, it follows that {g(AB)} is a large subarc of {\mathbb T} that covers almost all of the circle. But the same argument applies to the arcs {g(BC)} and {g(CA)}, a contradiction.

No matter how large the enclosed disk is, a tight combover around it can force arbitrarily large values of the Lipschitz constant of {f^{-1}}. Sad!

What about sufficient conditions?

  1. It is sufficient for the curve to be star-shaped with respect to the origin, in addition to enclosing the unit disk. (Equivalent statement: {\Gamma} has a polar equation {r = r(\theta)} in which {r \ge 1}.) Indeed, the nearest-point projection onto a convex set is a nonexpanding map, and projecting {\Gamma} onto the unit circle in this way gives the desired parameterization.
  2. It is sufficient for the curve to have curvature bounded by 1. I am not going into this further, because second derivatives should not be needed to control a Lipschitz constant.

Recall that for nonexpanding parameterizations, the length of the curve was a single quantity that mostly answered the existence question (length at most 4 — yes; length greater than {2\pi} — no). I do not know of such a quantity for the noncontracting case.

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