The tennis ball theorem states that a smooth-enough curve that bisects the surface area of a sphere must have at least four inflection points. There are plenty of sources on this but most of them are sloppy about what "smooth enough" means. One source that isn't sloppy about this is Thorbergsson and Umehara (1999), "A unified approach to the four vertex theorems II", which says that it is enough that the curve be $C^2$. But the usual description of the seam of a tennis ball (four semicircles meeting at points of mutual tangency) is only $C^1$, so their statement of the theorem does not apply to tennis balls.

In order to define a unique tangent great circle at each point of the curve, $C^1$ is enough. Defining inflection points from changes of sign of curvature (as done e.g. by Weiner (1977), "Global properties of spherical curves", whose Theorem 2 is a pre-Arnold instance of the tennis ball theorem) would instead need $C^2$, but Thorbergsson and Umehara give a careful definition of inflection points that appears to need only $C^1$. (Intersect the curve with the tangent great circle at a given point, find the connected component of the intersection that contains the given point, and then ask whether every neighborhood of the component has points of the curve on both sides of the great circle.) Is it possible to prove this theorem for $C^1$ curves?

Actually, I have in mind a proof that I think does work for $C^1$: the one from Angenent (1999), "Inflection points, extatic points and curve shortening". Angenent argues that the curve-shortening flow preserves smoothness and area-bisection, never increases the number of inflection points, and eventually converges to a great circle. When it does, you can look at the Fourier coefficients of the curve; the first one is zero because if it were nonzero we would be converging to a different great circle, so (based on an old result of Sturm) there are at least four inflections. Therefore the original curve must also have at least four inflections. Angenent is not very careful about his smoothness assumptions, but curve-shortening is known to instantaneously make the curve infinitely differentiable, so it seems to me that this argument should work even if we start with $C^1$. Does it?

But what I really want to know is not so much, does the theorem hold for $C^1$ curves, but rather, is there a publication I can point to that says it holds for $C^1$ curves? That's what I would need to be able to change the $C^2$ to $C^1$ in the Wikipedia article, which is my goal in asking this here.


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