Equichordal point problem

An equichordal curve is a closed planar curve for which a point in the plane exists such that all chords passing through this point are equal in length.{{citation

|author=Steven G. Krantz

|title=Techniques of Problem Solving

|publisher=American Mathematical Society

|year=1997

|isbn=978-0-8218-0619-7

}} Such a point is called an equichordal point. It is easy to construct equichordal curves with a single equichordal point, particularly when the curves are symmetric;{{citation

|author=Ferenc Adorján

|title=Equichordal curves and their applications – the geometry of a pulsation-free pump

|date=18 March 1999

|url=http://web.axelero.hu/fadorjan/gepfdp/geopfdp.pdf

}} the simplest construction is a circle.

It has long only been conjectured that no convex equichordal curve with two equichordal points can exist. More generally, it was asked whether there exists a Jordan curve $C$ with two equichordal points $O\_1$ and $O\_2$, such that the curve

$C$ would be star-shaped with respect to each of the two points.

Many results on equichordal curves refer to their excentricity. It turns out that the smaller the excentricity, the harder it is to disprove the existence of curves with two equichordal points. It can be shown rigorously that a small excentricity means that the curve must be close to the circle.

Let $C$ be the hypothetical convex curve with two equichordal points $O\_1$ and $O\_2$. Let $L$ be the common length of all chords of the curve $C$ passing through $O\_1$ or $O\_2$. Then excentricity is the ratio

:$a\; =\; \backslash frac\{\backslash |O\_1-O\_2\backslash |\}\{L\}$

where $\backslash |O\_1-O\_2\backslash |$ is the distance between the points $O\_1$ and $O\_2$.

The problem has been extensively studied, with significant papers published over eight decades preceding its solution:

1. In 1916, FujiwaraM. Fujiwara. Über die Mittelkurve zweier geschlossenen konvexen Curven in Bezug auf einen Punkt. Tôhoku Math J., 10:99–103, 1916 proved that no convex curves with three equichordal points exist.
2. In 1917, Blaschke, Rothe and Weitzenböck formulated the problem again.
3. In 1923, Süss showed certain symmetries and uniqueness of the curve, if it existed.
4. In 1953, G. A. Dirac showed some explicit bounds on the curve, if it existed.
5. In 1958, WirsingE. Wirsing, Zur Analytisität von Doppelspeichkurven, Arch. Math. 9 (1958), 300–307. showed that the curve, if it exists, must be an analytic curve. In this deep paper, he correctly identified the problem as perturbation problem beyond all orders.
6. In 1966, EhrhartR. Ehrhart, Un ovale à deux points isocordes?, Enseignement Math. 13 (1967), 119–124 proved that there are no equichordal curves with excentricities > 0.5.
7. In 1974, HallstromA. Hallstrom, Equichordal and Equireciprocal Points, Bogazici Univesitesi Dergisi Temel Bilimier-Sciences (1974), 83-88 gave a condition on the curve, if it exists, that shows it must be unique, analytic, symmetric and provides a means (given enough computer power) to demonstrate non-existence for any specific eccentricity.
8. In 1988, Michelacci proved that there are no equichordal curves with excentricities > 0.33. The proof is mildly computer-assisted.
9. In 1992, Schäfke and VolkmerR. Schäfke and H. Volkmer, Asymptotic analysis of the equichordal problem, J. Reine Angew. Math. 425 (1992), 9–60 showed that there is at most a finite number of values of excentricity for which the curve may exist. They outlined a feasible strategy for a computer-assisted proof. Their method consists of obtaining extremely accurate approximations to the hypothetical curve.
10. In 1996, Rychlik fully solved the problem.

Marek Rychlik's proof was published in the hard to read article.

There is also an easy to read, freely available on-line, research announcement article,Marek Rychlik, The Equichordal Point Problem, Electronic Research Announcements of the AMS,

1996, pages 108–123, available on-line at [https://www.ams.org/journals/era/1996-02-03/S1079-6762-96-00015-7/S1079-6762-96-00015-7.pdf] but it only hints at the ideas used in the proof.

The proof does not use a computer. Instead, it introduces a complexification of the original problem, and develops a generalization of the theory of normally hyperbolic invariant curves and stable manifolds to multi-valued maps $F:\backslash mathbb\{C\}^2\backslash to\backslash mathbb\{C\}^2$. This method allows the use of global methods of complex analysis. The prototypical global theorem is the Liouville's theorem. Another global theorem is Chow's theorem. The global method was used in the proof of Ushiki's Theorem.S. Ushiki. Sur les liaisons-cols des systèmes dynamiques analytiques. C. R. Acad. Sci. Paris, 291(7):447–449, 1980

Similar problems and their generalizations have also been studied.

1. The equireciprocal point problem
2. The general chordal problem of Gardner
3. Equiproduct point problem

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Category:Convex geometry

Category:Dynamical systems