A simple closed curve on the plane can be parameterized by a homeomorphism 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: for all . Note that Euclidean distance is used here, not arclength. And we still want to be a homeomorphism. Its noncontracting property simply means that the inverse 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:
- The curve must have length at least .
- The curve must have diameter at least 2.
- The curve must enclose some disk of radius 1. (Apply Kirszbraun’s theorem to 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”:
Indeed, suppose that is a nonexpanding map from this curve onto the unit circle. Let 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 is a large subarc of that covers almost all of the circle. But the same argument applies to the arcs and , 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 . Sad!
What about sufficient conditions?
- It is sufficient for the curve to be star-shaped with respect to the origin, in addition to enclosing the unit disk. (Equivalent statement: has a polar equation in which .) Indeed, the nearest-point projection onto a convex set is a nonexpanding map, and projecting onto the unit circle in this way gives the desired parameterization.
- 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 — no). I do not know of such a quantity for the noncontracting case.