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The Equichordal Point Problem
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## The Equichordal Point Problem

### Marek Rychlik

**Abstract.**
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*

**Retrieve entire article **

#### 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.

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