A point P inside a closed convex curve C is
called equichordal if every chord of C drawn through point P has the same
length. Is there a curve with two equichordal points?

It has been conjectured for a long time that a
curve with two equichordal points cannot exist. But let us suppose for a moment that such
a curve does exist. We will call this curve an equichordal curve. The distance
between the potential two equichordal points is called the excentricity of the
equichordal curve and will be denoted by a.
If such a curve existed then it would be a union of the two curves depicted in the
following motion picture for excentricities varying in the range [.8,.4]:
The two curves are invariant under a map T of
the plane into itself defined by the following rule in terms of the two potential
equichordal points O_{1} and O_{2}:
If you are at point P, walk in the direction of O_{1}
by distance 1; when you are done, walk in the direction of O_{2}
by distance 1. The final point Q of your trip is the image of P
under T, i.e. Q=T(P).
If the coordinates are chosen so that O_{1}=(b,0) and O_{2}=(b,0),
where b is a half of the excentricity (as in the above movie) then the two
curves are unique analytic curves invariant under T and passing throuh points (1/2,0)
and (1/2,0) respectively, and not equal to the xaxis (which is also
invariant).
Although numerical studies indicate clearly that the union of these two curves is not a
Jordan curve, the solution of the Equichordal Point Problem for all excentricities in the
range (0,1) takes about 70 journal pages.
It proves that the problem reduces to studying properties of the following functional
equation:
1/( f(m z)  f(z) ) + 1/( f(z)  f(z/m) ) = (1/a) (1 + f(z)^2)^(1/2)
for a complex function f(z). Here m is a parameter determined from the
equation m=(1+a)/(1a)
This function should have a simple pole at 0 and it is multivalued. 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)).