Equichordal point problem

In Euclidean plane geometry, the equichordal point problem is the question whether a closed planar convex body can have two equichordal points. The problem was originally posed in 1916 by Fujiwara and in 1917 by Wilhelm Blaschke, Hermann Rothe, and Roland Weitzenböck. A generalization of this problem statement was answered in the negative in 1997 by Marek R. Rychlik.
Problem statement.
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. Such a point is called an equichordal point. It is easy to construct equichordal curves with a single equichordal point, particularly when the curves are symmetric; the simplest construction is a circle.
It has long only been conjectured that no convex equichordal curve with two equichordal points can exist. More generally, it was asked whether there exists a Jordan curve formula_1 with two equichordal points formula_2 and formula_3, such that the curve 
formula_1 would be star-shaped with respect to each of the two points.
Excentricity (or eccentricity).
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.
Let formula_1 be the hypothetical convex curve with two equichordal points formula_2 and formula_3. Let formula_8 be the common length of all chords of the curve formula_1 passing through formula_2 or formula_3. Then excentricity is the ratio
where formula_13 is the distance between the points formula_2 and formula_3.
The history of the problem.
The problem has been extensively studied, with significant papers published over eight decades preceding its solution:
Rychlik's proof.
Marek Rychlik's proof was published in the hard to read article.
There is also an easy to read, freely available on-line, research announcement article, but it only hints at the ideas used in the proof.
The proof does not use a computer. 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 formula_16. This method allows the use of global methods of complex analysis. The prototypical global theorem is the Liouville's theorem. Another global theorem is Chow's theorem. The global method was used in the proof of Ushiki's Theorem.
See also.
Similar problems and their generalizations have also been studied.