Equichordal point problem

In Euclidean plane geometry, the equichordal point problem is the question whether a closed planar convex body can have two equichordal points.^[1] The problem was originally posed in 1916 by Fujiwara and in 1917 by Wilhelm Blaschke, Rothe and Weizenböck.^[2] A generalization of this problem statement was answered in the negative in 1997 by Marek R. Rychlik.^[3]

Problem statement

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.^[4] Such a point is called an equichordal point. It is easy to construct equichordal curves with a single equichordal point,^[4] particularly when the curves are symmetric;^[5] 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 O₁ and O₂, such that the curve C would be star-shaped with respect to each of the two points.^[1][3]

Excentricity (or eccentricity)

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.^[6]

Let C be the hypothetical convex curve with two equichordal points O₁ and O₂. Let L be the common length of all chords of the curve C passing through O₁ or O₂. Then excentricity is the ratio

$a = \frac{\|O_1-O_2\|}{L}$

where $\|O_1-O_2\|$ is the distance between the points O₁ and O₂.

The history of the problem

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

1. In 1916 Fujiwara^[7] proved that no convex curves with three equichordal points exist.
2. In 1917 Blaschke, Rothe and Weitzenböck^[2] 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 Wirsing^[8] 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 Ehrhart^[9] proved that there are no equichordal curves with excentricities > 0.5.
7. In 1988 Michelacci proved that there are no equichordal curves with excentricities > .33. The proof is mildly computer-assisted.
8. In 1992 Shäfke and Volkmer^[6] 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 methods consists in obtaining extremaly accurate approximations to the hypothetical curve.
9. In 1996 Rychlik^[3] fully solved the problem.

Rychlik's proof

Marek Rychlik's proof was published in the hard to read article.^[3] There is also an easy to read, freely available on-line, research announcement article,^[10] 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 $F:\mathbb{C}^2\to\mathbb{C}^2$. 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.^[11]

See also

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

References

1. ^ ^{a b} Victor Klee; Stan Wagon, Old and New Unsolved Problems in Plane Geometry and Number Theory, Mathematical Association of America, doi:10.2277/0883853159, ISBN 9780883853153 
2. ^ ^{a b} W. Blaschke, W. Rothe, and R. Weitzenböck. Aufgabe 552. Arch. Math. Phys., 27:82, 1917
3. ^ ^{a b c d} Marek R. Rychlik (1997), "A complete solution to the equichordal point problem of Fujiwara, Blaschke, Rothe and Weizenböck", Inventiones Mathematicae 129 (1): 141–212 
4. ^ ^{a b} Steven G. Krantz (1997), Techniques of Problem Solving, American Mathematical Society, ISBN 978-0-8218-0619-7 
5. ^ Ferenc Adorján (18 March 1999), Equichordal curves and their applications – the geometry of a pulsation-free pump, http://web.axelero.hu/fadorjan/gepfdp/geopfdp.pdf 
6. ^ ^{a b} R. Schäfke and H. Volkmer, Asymptotic analysis of the equichordal problem, J. Reine Angew. Math. 425 (1992), 9–60
7. ^ M. Fujiwara. Über die Mittelkurve zweier geschlossenen konvexen Curven in Bezug auf einen Punkt. Tôhoku Math J., 10:99–103, 1916
8. ^ E. Wirsing, Zur Analytisität von Doppelspeichkurven, Arch. Math. 9 (1958), 300–307.
9. ^ R. Ehrhart, Un ovale à deux points isocordes?, Enseignement Math. 13 (1967), 119–124
10. ^ Marek Rychlik, The Equichordal Point Problem, Electronic Research Announcements of the AMS, 1996, pages 108–123, available on-line at [1]
11. ^ S. Ushiki. Sur les liaisons-cols des systèmes dynamiques analytiques. C. R. Acad. Sci. Paris, 291(7):447–449, 1980