## Introduction

### The beginnings

The equichordal point problem was posed in 1916 by Fujiwara and in the subsequent article by Blaschke, Rothe and Weitzenböck in 1917. The formulation sounds like a puzzle in a geometry textbook. It starts with the following definition:

A point inside a closed, planar curve is called an equichordal point if every chord passing through that point has the same length.

For example, the center of a circle is equichordal, as every chord passing through the center has length equal to the circle diameter. One should note that a chord is simply a segment connecting two points on the curve.

Fujiwara himself observed that there are many curves, other than a circle, that have at least one equichordal point. He also proved that there is no curve with three equichordal points.

After the work of Fujiwara, the only open case left was that of the two equichordal points. This is essentially the equichordal point problem. There are some problems with interpreting the problem for an arbitrary simple closed curve (i.e. a curve without intersections). The notion of convexity was used to resolve these issues. We recall that a curve is convex if it is a boundary of a 2-dimensional region which is convex. A region is convex if every two points in the region can be connected with a line segment which is totally contained in the region.

The Equichordal Point Problem: Is there a convex planar curve with two equichordal points?

## The Equichordal Point Problem entry in the MathWorld On-Line Encyclopedia

Thanks to the excellent work of Eric Weinstein, the Equichordal Point Problem has an entry in the encyclopedia. Here is the link:
http://mathworld.wolfram.com/EquichordalPointProblem.html

## An animation

The animation below illustrates a part of the method of the proof. When one tries to construct a curve with two equichordal points positioned at a distance $a$ (called the excentricity, or eccentricity), with the length of the chords equal to 1, one arrives at two approximating curves, grown from points (-0.5,0) and (0.5,0) by an iterative process. In fact, the curves are reflections of each other in the $y$-axis and are symmetric with respect to reflections in the $x$-axis.

If for some value of $a$ the curves coincided then woud have a curve with two equichordal points. However, the curves oscillate, which prevents them from coinciding. The construction of the above animation involves complicated calculations of series expansions with a CAS, followed by a numerical iteration process.

The method of Shafke and Volkmer and an earlier work of Wirsing focus on studying the size of the oscillations of the potential solutions to the equichordal point problem. It turns out that they die down with $a$ faster than any power of $a$. In addition, one can construct special asymptotic expansions which capture the size of the oscillations for small $a$.

## The solution of a functional equation in the complex domain

It proves that the problem reduces to studying properties of the following functional equation:

$$\frac{1}{f(\mu z) - f(z) } + \frac{1}{ f(z) - f(z/\mu) } = \frac{1}{a\sqrt{1 + f(z)^2}}$$

for a complex function $f(z)$. Here $\mu$ is a parameter determined from the equation

$\displaystyle{\mu=\frac{1+a}{1-a}.}$

This function should have a simple pole at $0$ and it is multi-valued. Its meromorphic continuation can be found from the above equation. The following picture is obtained if one plots $|f(z)|$. The color is determined by $\Re f(z)$.