Classify minimizers at the distinguished lengths

Classify all simple closed curves on the unit sphere that minimize the mean geodesic distance to the curve among curves of length exactly L_n=2π/sin(π/(2n)), for each positive integer n.

Background

For each positive integer n, the paper proves that the minimum mean distance among curves of length at most L_n equals j(L_n)=π/(2n)−tan(π/(4n)), and that the sphere-filling ropes of Gerlach and von der Mosel attain this value. It also proves structural properties shared by every minimizer: such a curve has length exactly L_n, bisects the sphere into two disks of area 2π, and has inward collars of maximal possible area at every depth.

The paper does not determine whether the known sphere-filling ropes exhaust all minimizers at L_n. The unresolved problem is therefore to classify every minimizing curve at each distinguished length, without imposing the additional positive-thickness hypothesis under which a classification is obtained.

References

Classify all minimizers at L=L_n, and determine J(L) for L>2π with L\notin{L_n}.

— The mean distance to a simple closed curve on the sphere  (2609.09638 - Pimentel, 9 Sep 2026) in Question environment, Section 6, “Other lengths”

Whether the bound $j(L)$ is attained at such an $L$ is open: by Proposition~\ref{prop:equality} a curve of that length with $\mathcal{J}=j(L)$ would bisect $S$ into two disks of inradius $\vartheta(L)$ and would attain the covering bound of Corollary~\ref{cor:cover} at the radius $\vartheta(L)\notin{\vartheta_{n}}$, which Proposition~\ref{prop:classify} rules out for curves of thickness at least $\sin\vartheta$ but not in general.

— The mean distance to a simple closed curve on the sphere  (2609.09638 - Pimentel, 9 Sep 2026) in Question environment, Section 6, “Other lengths”