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The Equichordal Point Problem. (English) Zbl 0871.52001
Summary: 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.
52A10 Convex sets in \(2\) dimensions (including convex curves)
37F99 Dynamical systems over complex numbers
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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