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The Equichordal Point Problem


Author: Marek Rychlik
Journal: Electron. Res. Announc. Amer. Math. Soc. 2 (1996), 108-123
MSC (1991): Primary 52A10, 39A; Secondary 39B, 58F23, 30D05
DOI: https://doi.org/10.1090/S1079-6762-96-00015-7
MathSciNet review: 1426720
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Abstract: If $C$ is a Jordan curve on the plane and $P, Q\in C$, then the segment $\overline {PQ}$ is called a chord of the curve $C$. A point inside the curve is called equichordal if every two chords through this point have the same length. Fujiwara in 1916 and independently Blaschke, Rothe and Weitzenböck in 1917 asked whether there exists a curve with two distinct equichordal points $O_1$ and $O_2$. This problem has been fully solved in the negative by the author of this announcement just recently. The proof (published elsewhere) reduces the question to that of existence of heteroclinic connections for multi-valued, algebraic mappings. In the current paper we outline the methods used in the course of the proof, discuss their further applications and formulate new problems.


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Additional Information

Marek Rychlik
Affiliation: Department of Mathematics, University of Arizona, Tucson, AZ 85721
Email: rychlik@math.arizona.edu

DOI: https://doi.org/10.1090/S1079-6762-96-00015-7
Keywords: Equichordal, heteroclinic, convex, multi-valued
Received by editor(s): September 15, 1996
Additional Notes: This research has been supported in part by the National Science Foundation under grant no. DMS 9404419.
Communicated by: Krystyna Kuperberg
Article copyright: © Copyright 1997 Marek Rychlik