Establish existence of minimizers at fixed length

Determine whether, for each fixed L>0, there exists a simple closed curve on the unit sphere of length exactly L that attains the infimum J(L) of the mean geodesic distance.

Background

The functional assigning mean geodesic distance to a curve is continuous with respect to Hausdorff distance, but the class of simple closed curves having exactly a prescribed length is not closed in the Hausdorff metric. Length is only lower semicontinuous under the relevant convergence.

Consequently, the paper cannot conclude that the infimum J(L) is attained for an arbitrary fixed length. It explicitly identifies the existence of a minimizer as an unresolved problem, independently of the unresolved classification and exact-value questions.

References

Even the existence of a minimizer at a fixed $L$ is open, since ${\mathcal{C}:\length(\mathcal{C})=L}$ is not closed for $d_{H}$ and length is only lower semicontinuous.

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