Curve Closest to Sphere

We propose a solution to the tenth of Professor Clark Kimberling's unsolved problems found on https://faculty.evansville.edu/ck6/integer/unsolved.html. We are required to find the parametric equations of a simple and closed curve $C$ on the unit sphere $S$ with arc-length $4 \pi$, that minimizes the mean arc-distance from $S$ to $C$. We give explicit definitions of the mean arc-distance from $C$ to $S$, $M$ and the mean arc-distance from $S$ to $C$, $\tilde{M}$. We show that these two quantities are not the same. We show that for all closed and simple curves $C$ of arc-length $4 \pi$ on $S$, $M$ is constant and is equal to $2 \pi^{2}$. Therefore all such curves minimize $M$. We show that in contrast, $\tilde{M}$ varies for different closed and simple curves $C$ of arc-length $4 \pi$ on $S$. We find such a curve that minimizes $\tilde{M}$.