Outdated Archival Version

These pages are not updated anymore. They reflect the state of 20 August 2005. For the current production of this journal, please refer to http://www.math.psu.edu/era/.

This journal is archived by the American Mathematical Society. The master copy is available at http://www.ams.org/era/

If $C$ is a Jordan curve on the plane and $P, Q\in C$, then the segment $\overline{PQ}$ is called a {\em chord} of the curve $C$. A point inside the curve is called {\em equichordal} if every two chords through this point have the same length. Fujiwara in 1916 and independently Blaschke, Rothe and Weitzenb\"ock 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.

Copyright 1997 Marek Rychlik

TeX source (49K)
DVI (67K)
PostScript (584K)

Article Info

• ERA Amer. Math. Soc. 02 (1996), pp.108-123
• Publisher Identifier: S 1079-6762(96)00015-7
• 1991 Mathematics Subject Classification. Primary 52A10, 39A; Secondary 39B, 58F23, 30D05
• Key words and phrases. Equichordal, heteroclinic, convex, multi-valued
• Received by the editors September 15, 1996
• Communicated by Krystyna Kuperberg
• Comments (When Available)

Marek Rychlik

Department of Mathematics, University of Arizona, Tucson, AZ 85721

E-mail address: rychlik@math.arizona.edu

This research has been supported in part by the National Science Foundation under grant no. DMS 9404419.