TITLE: Surprising properties of closed planar curves
QUESTION [25 upvotes]: In https://arxiv.org/abs/2002.05422 I proved with elementary topological methods that a smooth planar curve with total turning number a non-zero integer multiple of $2\pi$ (the tangent fully turns a non-zero number of times) can always be split into 3 arcs rearrangeable to a closed smooth curve. Here comes one example of what I called the 2-cut theorem.

I am now writing the introduction to my thesis and I would like to cite more examples of counterintuitive properties of planar closed curves like, I think, the one above. The inscribed square problem, asking whether every Jordan curve admits an inscribed square (pic from Jordan curves admitting only acyclic inscriptions of squares), alredy came to my mind.

Such a property is still a conjecture for the general case, but proofs have been provided for several special cases and the easier inscribed rectangle problem can be solved with a beautiful topological argument (3Blue1Brown made a very nice video about that https://www.youtube.com/watch?v=AmgkSdhK4K8&t=169s).
My question: what are other surprising/counterintuitive properties of closed planar curves you are aware of?
Note: this question was originally posted on MathStackExchange since I thought it was suitable for a more general audience. Apparently, it requires a slightly broader knowledge of literature.

REPLY [3 votes]: How about the ``Oval's Problem" of Benguria and Loss.  This is a (somewhat surprisingly) open problem connecting spectral theory with plane geometry.  The conjecture is that for any simple closed curve $\sigma$ of length $2\pi$ and any periodic function $f:[0,2\pi]\to \mathbb{R}$ one has
$$
\int_0^{2\pi} |f'(s)|^2+\kappa(s)^2 |f(s)|^2 ds \geq \gamma \int_{0}^{2\pi} |f(s)|^2 ds
$$
for $\gamma=1$.  Here $s$ is the arclength parameter and $\kappa(s)$ is the geodesic curvature. In other words, the lowest eigenvalue of $-\frac{d^2}{ds^2}+\kappa(s)^2$ is bounded from below by $1$ on a closed curve.  This lowest bound is achieved by the circle, but is actually achieved by a whole family of "ovals" that deteriorate to a multiplicity two line segment.  Interestingly, this problem is related the sharp Lieb-Thirring inequality which is a purely spectral theoretic problem (this was Benguria and Loss's motivation).  It is also related to minimal surface theory.
In their paper, Benguria and Loss show that this inequality holds with $\gamma=\frac{1}{2}$.  This is actually, sharp if one expands the set of curves to include those that total turning number 1 so one has to use the curve is closed in some way (and not just that $\int_0^{2\pi} k(s)ds=2\pi$.  Various other authors have worked on parts of this problem: Burchard and Thomas showed the ovals where local minima of the lowest eigenvalue (so the problem is solved near curves in the family, Linde showed closed convex curves have $\gamma\geq 0.6085$, Denzler showed there is a closed convex curve that minimized the value of $\gamma$ and Bernstein and Mettler discussed the symmetry of the problem (related to projective geometry) and showed some weaker geometric inequalities held for closed convex curves (but not for curves of turning number 1).