A complete solution to the equichordal point problem of Fujiwara, Blaschke, Rothe and Weizenböck

Abstract.

The Equichordal Point Problem can be formulated in simple geometric terms. 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. The question was whether there exists a curve with two distinct equichordal points $O_1$ and $O_2$. The problem was posed by Fujiwara in 1916 and independently by Blaschke, Rothe and Weizenböck in 1917, and since then it has been attacked by many mathematicians. In the current paper we prove that if $O_1$ and $O_2$ are two distinct points on the plane and $C$ is a Jordan curve such that the bounded region $D$ cut out by $C$ is star-shaped with respect to both $O_1$ and $O_2$ then $C$ is not equichordal. The original question was posed for convex $C$, and thus we have solved the Equichordal Point Problem completely. Our method is based on the observation that $C$ would be an invariant curve for an algebraic map of the plane. It would also form a heteroclinic connection. We complexify the map and obtain a multivalued algebraic map of ${\Bbb C}^2$. We develop criteria for the existence of heteroclinic connections for such maps.