Equichordal point problem: Difference between revisions

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In geometry, Rychlik's theorem, named after Marek R. Rychlik, solves the equichordal point problem of Fujiwara, originally posed in 1916. The problem was rediscovered in 1917 by Wilhelm Blaschke, Rothe and Weizenböck^[1].

The theorem

There is no $C$ be a Jordan curve 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.

In particular, there is no convex and closed curve with two equichordal points, which is the original formulation of the equichordal point problem.

The published proof

The proof was published in the hard to read article ^[2]. There is also an easy to read, freely available on-line, research announcement article ^[3], but it only hints at the ideas used in the proof.

Definition of excentricity (or eccentricity)

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={\frac {\|O_{1}-O_{2}\|}{L}}

where $\|O_{1}-O_{2}\|$ is the distance between the points $O_{1}$ and $O_{2}$ .

Many results on equichordal points refer to the excentricity. It turns out that the smaller the $a$ , the harder is to disprove the existence of the curves with two equichordal points. It can be shown rigorously, that small excentricity means that the curve must be close to the circle^[4].

The history of the problem

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

In 1916 Fujiwara^[5] proved that no convex curves with three equichordal points exist.
In 1917 Blaschke, Rothe and Weitzenböck^[1] formulated the problem again.
In 1923 Süss showed certain symmetries and uniqueness of the curve, if it existed.
In 1953 G. A. Dirac showed some explicit bounds on the curve, if it existed.
In 1958 Wirsing^[6] 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.
In 1966 Ehrhart^[7] proved that there are no equichordal curves with excentricities > 0.5.
In 1988 Michelacci proved that there are no equichordal curves with excentricities > .33. The proof is mildly computer-assisted.
In 1992 Shäfke and Volkmer^[4] 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.
In 1996 Rychlik^[2] fully solved the problem. The proof does not use a computer. Instead it introduces complexification of the original problem, and develops a generalization of the theory of normally hyperbolic invariant curves and stable manifolds, which allow 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^[8].

References

^ ^a ^b W. Blaschke, W. Rothe, and R. Weitzenböck. Aufgabe 552. Arch. Math. Phys., 27:82, 1917
^ ^a ^b Marek R. Rychlik, A complete solution to the equichordal point problem of Fujiwara, Blaschke, Rothe and Weizenböck, Inventiones Mathematicae, 1997, Volume 129, Number 1, pages 141–212
^ Marek Rychlik, The Equichordal Point Problem, Electronic Research Announcements of the AMS, 1996, pages 108–123, available on-line at [1]
^ ^a ^b R. Schäfke and H. Volkmer, Asymptotic analysis of the equichordal problem, J. Reine Angew. Math. 425 (1992), 9–60
^ M. Fujiwara. Über die Mittelkurve zweier geschlossenen konvexen Curven in Bezug auf einen Punkt. Tôhoku Math J., 10:99–103, 1916
^ E. Wirsing, Zur Analytisität von Doppelspeichkurven, Arch. Math. 9 (1958), 300–307.
^ R. Ehrhart, Un ovale à deux points isocordes?, Enseignement Math. 13 (1967), 119–124
^ S. Ushiki. Sur les liasons-cols des systemes dynamiques analytiques. C. R. Acad. Sci. Paris, 291(7):447-449, 1980

[Blaschke-1] W. Blaschke, W. Rothe, and R. Weitzenböck. Aufgabe 552. Arch. Math. Phys., 27:82, 1917

[The_Proof-2] Marek R. Rychlik, A complete solution to the equichordal point problem of Fujiwara, Blaschke, Rothe and Weizenböck, Inventiones Mathematicae, 1997, Volume 129, Number 1, pages 141–212

[The_Announcement-3] Marek Rychlik, The Equichordal Point Problem, Electronic Research Announcements of the AMS, 1996, pages 108–123, available on-line at [1]

[Volkmer-4] R. Schäfke and H. Volkmer, Asymptotic analysis of the equichordal problem, J. Reine Angew. Math. 425 (1992), 9–60

[Fujiwara-5] M. Fujiwara. Über die Mittelkurve zweier geschlossenen konvexen Curven in Bezug auf einen Punkt. Tôhoku Math J., 10:99–103, 1916

[Wirsing-6] E. Wirsing, Zur Analytisität von Doppelspeichkurven, Arch. Math. 9 (1958), 300–307.

[Erhhart-7] R. Ehrhart, Un ovale à deux points isocordes?, Enseignement Math. 13 (1967), 119–124

[Ushiki_Standard_Map_Theorem-8] S. Ushiki. Sur les liasons-cols des systemes dynamiques analytiques. C. R. Acad. Sci. Paris, 291(7):447-449, 1980

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@@ Line 29: / Line 29: @@
 # In 1988 Michelacci proved that  there are no equichordal curves with excentricities > .33. The proof is mildly computer-assisted.
 # In 1992 [[Shäfke]] and [[Volkmer]]<ref name="Volkmer">R. Schäfke and H. Volkmer, Asymptotic analysis of the equichordal problem, J. Reine Angew. Math. 425 (1992), 9–60</ref> 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.
-# In 1996 Rychlik<ref name="The_Proof"/> fully solved the problem. The proof does not use a computer. Instead it introduces [[complexification]] of the original problem, and develops a generalization of the theory of [[Normally hyperbolic invariant manifold|normally hyperbolic invariant curves]] and [[stable manifold]]s, which allow the use of global methods of [[complex analysis]]. The prototypical global theorem is the [[Liouville's theorem (complex analysis) | Liouville's theorem]]. Another global theorem is [[Algebraic geometry and analytic geometry|Chow's theorem]].
+# In 1996 Rychlik<ref name="The_Proof"/> fully solved the problem. The proof does not use a computer. Instead it introduces [[complexification]] of the original problem, and develops a generalization of the theory of [[Normally hyperbolic invariant manifold|normally hyperbolic invariant curves]] and [[stable manifold]]s, which allow the use of global methods of [[complex analysis]]. The prototypical global theorem is the [[Liouville's theorem (complex analysis) | Liouville's theorem]]. Another global theorem is [[Algebraic geometry and analytic geometry|Chow's theorem]]. The global method was used in the proof of [[Ushiki's Theorem]]<ref name="Ushiki_Standard_Map_Theorem">S. Ushiki. Sur les liasons-cols des systemes dynamiques analytiques. C. R. Acad. Sci. Paris, 291(7):447-449, 1980</ref>.
 == See also ==