The mean distance to a simple closed curve on the sphere

Kimberling’s Problem 10 asks for a simple closed curve of prescribed length $L$ (in particular, $L=4\pi$) on the unit sphere minimizing the mean geodesic distance $\mathcal{J}$ from a point of the sphere to the curve. For a positive integer $n$, put ${\vartheta }_n=\pi \mathrm{/}(2n)$ and $L_n=2\pi \mathrm{/}\sin {\vartheta }_n$. We show that the minimum of $\mathcal{J}$ over rectifiable simple closed curves of length at most $L_n$ equals ${\vartheta }_n-\tan ({\vartheta }_n\mathrm{/}2)$, that it is attained only by curves of length exactly $L_n$, and that the sphere-filling ropes ${\beta }^{n,k}$ of Gerlach and von der Mosel attain it. Kimberling’s case is $n=3$: at $L=4\pi$ the minimum is $\pi \mathrm{/}6+\sqrt{3}-2=0.255649\dots$, attained by an explicit six-arc curve and by its mirror image. For $L\le 2\pi$ we determine $J(L)$, the infimum of $\mathcal{J}$ over curves of length $L$, exactly: it equals $\pi \mathrm{/}2-L\mathrm{/}(2\pi )$, attained precisely by the circles of length $L$. At the lengths $L_n$ we do not classify all minimizers, but show that every one of them bisects the sphere into two disks of area $2\pi$ and inradius ${\vartheta }_n$ whose inward collars have the largest possible area at every depth. The great circle is the only minimizer for $n=1$, and the ${\beta }^{n,k}$ are, up to congruence, the only ones of thickness at least $\sin {\vartheta }_n$. For arbitrary $L$ the function $J$ is nonincreasing, and together with the above this brackets it between two explicit values.