What Jordan curve of length $1$ maximizes the expected length of a chord between two uniformly picked points on it? For a (sufficiently nice) Jordan curve $\sigma :[0,1] \to \mathbb{R}^2$ with unit length and natural (w.r.t. arc length) parametrization, we denote the expected value of the distance between two points that are picked from the path independently and uniformly by $c(\sigma)$. So

$$c(\sigma) = \mathbb{E}[ \space|\sigma(T_1)- \sigma(T_2)| \space],$$

where $T_1, T_2 \sim U([0,1])$ are independent.


  What path maximizes $c$?


For example, for $\sigma =$ circle (of radius $\frac{1}{2\pi}$ to have unit perimeter),
$$c(\sigma) = \frac{2}{\pi^2}$$

(It is known that for unit circle we get $\frac{4}{\pi}$ so scaling leads to the value $\frac{2}{\pi^2}$.)
Is circle the maximizing path?

PS. Here is a small application for testing different kinds of paths. The circle seemed to be the best among all paths I tried out.
I'm pretty certain that a straight line segment is the optimizing path. 

For then the expected length appears to be $\frac{1}{3}$ (if my back-of-the-envelope computation is correct), which is much larger than $\frac{2}{\pi^2}$.
I'm pretty certain that a straight line segment is the optimizing path. 

For then the expected length appears to be $\frac{1}{3}$ (if my back-of-the-envelope computation is correct), which is much larger than $\frac{2}{\pi^2}$.