In Euclidean plane geometry, the equichordal point problem is the question whether a closed planar convex body can have two equichordal points.
[1] The problem was originally posed in 1916 by Fujiwara and in 1917 by Wilhelm Blaschke, Hermann Rothe, and Roland Weitzenböck.
[2] A generalization of this problem statement was answered in the negative in 1997 by Marek R.
[3] An equichordal curve is a closed planar curve for which a point in the plane exists such that all chords passing through this point are equal in length.
It is easy to construct equichordal curves with a single equichordal point,[4] particularly when the curves are symmetric;[5] the simplest construction is a circle.
More generally, it was asked whether there exists a Jordan curve
[1][3] Many results on equichordal curves refer to their excentricity.
It turns out that the smaller the excentricity, the harder it is to disprove the existence of curves with two equichordal points.
It can be shown rigorously that a small excentricity means that the curve must be close to the circle.
be the hypothetical convex curve with two equichordal points
The problem has been extensively studied, with significant papers published over eight decades preceding its solution: Marek Rychlik's proof was published in the hard to read article.
[3] There is also an easy to read, freely available on-line, research announcement article,[11] but it only hints at the ideas used in the proof.
Instead, it introduces a complexification of the original problem, and develops a generalization of the theory of normally hyperbolic invariant curves and stable manifolds to multi-valued maps
The global method was used in the proof of Ushiki's Theorem.