Equichordal point problem: Difference between revisions
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# In 1916 Fujiwara<ref name="Fujiwara">M.~Fujiwara. Über die Mittelkurve zweier geschlossenen konvexen Curven in |
# In 1916 Fujiwara<ref name="Fujiwara">M.~Fujiwara. Über die Mittelkurve zweier geschlossenen konvexen Curven in |
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Bezug auf einen Punkt. Tôhoku Math J., 10:99--103, 1916 </ref> proved that no convex curves with three equichordal points exist. |
Bezug auf einen Punkt. Tôhoku Math J., 10:99--103, 1916 </ref> proved that no convex curves with three equichordal points exist. |
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# In 1917 Blaschke, Rothe and Weitzenböck |
# In 1917 Blaschke, Rothe and Weitzenböck<ref>Blaschke</ref> formulated the problem again. |
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# In 1923 Süss showed certain symmetries and uniqueness of the curve, if it existed. |
# In 1923 Süss showed certain symmetries and uniqueness of the curve, if it existed. |
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# In 1953 G. A. Dirac showed some explicit bounds on the curve, if it existed. |
# In 1953 G. A. Dirac showed some explicit bounds on the curve, if it existed. |
Revision as of 07:35, 25 November 2010
The theorem 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 be a Jordan curve with two equichordal points, with respect to which the curve would be star-shaped. In particular, there is no convex and closed curve with two equichordal points.
The Published Proof
The proof was published in the hard to read article [2] There is also easy to read research announcement [3], but it only hints at the ideas used in the proof.
The History of the Problem
- In 1916 Fujiwara[4] proved that no convex curves with three equichordal points exist.
- In 1917 Blaschke, Rothe and Weitzenböck[5] 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 eccentricities > 0.5.
- In 1988 Michelacci proved that there are no equichordal curves with eccentricities > .33. The proof is mildly computer-assisted.
- In 1992 Shäfke and Volkmer[8] showed that there is at most a finite number of values of eccentricity for which the curve may exist. They outlined a feasible strategy for a computer-assisted proof.
- In 1996 Rychlik[2] fully solved the problem. The proof does not use a computer.
References
- ^ 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]
- ^ M.~Fujiwara. Über die Mittelkurve zweier geschlossenen konvexen Curven in Bezug auf einen Punkt. Tôhoku Math J., 10:99--103, 1916
- ^ Blaschke
- ^ 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
- ^ R. Schäfke and H. Volkmer, Asymptotic analysis of the equichordal problem, J. Reine Angew. Math. 425 (1992), 9-60