Surprising properties of closed planar curves

Question

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.

Answer 1

The Four Vertex Theorem, due to Mukhopadhyaya in 1909, states that a plane closed simple smooth curve with positive curvature has at least four vertices, where a vertex is a local maximum or minimum of the curvature. There's a detailed discussion in a chapter of Fuchs and Tabachnikov, Mathematical Omnibus.

Answer 2

Chakerian's Theorem (proved in this paper) that a closed curve of length L in the unit ball in $\mathbb{R}^n$ has total curvature at least L.

(In this later paper Chakerian gave a simpler proof and noted that equality holds iff the curve is of length $2\pi n$ and winds round the unit circle $n$ times.)

Answer 3

One result that I initially found a bit surprising is Grayson's theorem. It's a little bit of a different flavor than the other examples but I think it's interesting and worth a mention.

Given a closed planar curve $\gamma$ which is smooth enough ($C^2$ is sufficient but it's possible to be deal with less regular curves), there is a process known as curve shortening flow, which deforms the curve using the flow $$ \frac{\partial \gamma}{\partial t} = \kappa N. $$

Here, $\kappa$ is the unsigned curvature and $N$ is the unit normal vector. This is called curve shortening flow because it tends to shrink curves.

Grayson's theorem states that if you start with a closed curve that does not cross itself, then under the flow it will remain embedded and eventually become convex. Gage and Hamilton had earlier shown that a convex curve shrinks to a round point, so this shows that any embedded curve shrinks to a round point.

Grayson's original proof is pretty involved. Now there are more conceptual proofs of this result (Huisken has a particularly nice one), so it's a bit less surprising. Nonetheless, it's possible to draw some really crazy curves and somehow the flow avoids itself and makes the curve convex. A good resource for this is Anthony Carapetis' website, which has an applet demonstrating the flow. Note that if you allow the curve to cross itself, the theorem fails and you can get local kinks to appear.

Answer 4

I post as an answer to not overload the original entry and to contribute myself to the nice flow of results.

Browsing the book "Unsolved problems in geometry: Unsolved Problems in Intuitive Mathematics" by Croft, Falconer and Guy (after a good hint from the Reddit user Giovanni_Resta), the following properties of closed convex planar curves came out.

• Given any closed convex planar curve, there always exist three concurrent chords bisecting each other and cutting at prescribed angle. The reference given is "On chords of convex curves", H. Steinhaus - Bull. Acad. Polon. Sci. Cl. III, 1957, which I was not able to find online. The general problem for simple closed curve is apparently still open and very much related to the general inscribed square problem I already mentioned in the question.
• Given any closed convex planar curve, there exists a point in its interior which lies on 4 normals through 4 distinct boundary points. The theorem was proven by Heil also for 3D convex bodies and 6 normals and it is conjectured for convex bodies of $\mathbb{R}^d$ and $2d$ normals. The reference given is "Concurrent normals and critical points under weak smoothness assumptions", E. Heil - NYASA, 1985.

To those, I add a characterization for $C^3$ closed planar curves (not necessarily simple), which I found in "Robust fairing via conformal curvature" - K. Crane, U. Pinkall, and P. Schröder, ACM Trans. Graphics 32, 2013.

A $C^3$ planar curve $\gamma$ parametrized on $[a,b]$ and with curvature $k$ is closed iff $\int_a^b k'\gamma=0 $.

If it is not surprising, it is at least a weird relation between dependent descriptors.

Answer 5

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).