(Reformulation of this question)
I want to find which convex closed curve of length $2\pi$ has the smallest average distance of all its points to the origin. For instance, given a curve $\gamma(s)$ with arc-length parametrization, the average distance to the origin is given by:
$$\dfrac{1}{2\pi} \int_{0}^{2\pi} |\gamma| \, ds$$
So I need to minimize that integral over all $\gamma(s)$ that satisfy the condition of being: closed, convex, and with length $2\pi$.
Here is my approach:
First I calculated that integral for the family of ovals with curvature $\kappa(s)=1 /\left(a^{2} \cos ^{2}(s)+a^{-2} \sin ^{2}(s)\right)$, for several $a$. For $a=1$, it's just the circle with average distance to the origin of 1. As $a$ increases, the curve gets flatter and the value for the average distance decreases to a value of $\pi/4$, when $a\to \infty$.
So $\pi/4$ is my best lower bound so far, I tried using Euler-Lagrange equation but I'm not sure how many constraints to use.