The mean distance to a simple closed curve on the sphere
Abstract
Kimberling's Problem 10 asks for a simple closed curve of prescribed length L (in particular, L=4π) on the unit sphere minimizing the mean geodesic distance 𝒥 from a point of the sphere to the curve. For a positive integer n, put ϑn=π/(2n) and Ln=2π/sinϑn. We show that the minimum of 𝒥 over rectifiable simple closed curves of length at most Ln equals ϑn−tan(ϑn/2), that it is attained only by curves of length exactly Ln, and that the sphere-filling ropes βn,k of Gerlach and von der Mosel attain it. Kimberling's case is n=3: at L=4π the minimum is π/6+√3−2=0.255649…, attained by an explicit six-arc curve and by its mirror image. For L≤2π we determine J(L), the infimum of 𝒥 over curves of length L, exactly: it equals $π/2-L/(2π)$, attained precisely by the circles of length L. At the lengths Ln we do not classify all minimizers, but show that every one of them bisects the sphere into two disks of area 2π and inradius ϑn whose inward collars have the largest possible area at every depth. The great circle is the only minimizer for n=1, and the βn,k are, up to congruence, the only ones of thickness at least sinϑn. For arbitrary L the function J is nonincreasing, and together with the above this brackets it between two explicit values.
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BibTeX
@misc{pimentel2026mean,
title = {{The mean distance to a simple closed curve on the sphere}},
author = {J. V. M. Pimentel},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09638},
eprint = {2609.09638},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09638},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The mean distance to a simple closed curve on the sphere
AU - J. V. M. Pimentel
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.09638
UR - https://arxiv.org/abs/2609.09638
AB - Kimberling's Problem 10 asks for a simple closed curve of prescribed length L (in particular, L=4π) on the unit sphere minimizing the mean geodesic distance 𝒥 from a point of the sphere to the curve. For a positive integer n, put ϑ_n=π/(2n) and L_n=2π/sinϑ_n. We show that the minimum of 𝒥 over rectifiable simple closed curves of length at most L_n equals ϑ_n−tan(ϑ_n/2), that it is attained only by curves of length exactly L_n, and that the sphere-filling ropes β^n,k of Gerlach and von der Mosel attain it. Kimberling's case is n=3: at L=4π the minimum is π/6+√3−2=0.255649…, attained by an explicit six-arc curve and by its mirror image. For L≤2π we determine J(L), the infimum of 𝒥 over curves of length L, exactly: it equals $π/2-L/(2π)$, 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 2π and inradius ϑ_n whose inward collars have the largest possible area at every depth. The great circle is the only minimizer for n=1, and the β^n,k are, up to congruence, the only ones of thickness at least sinϑ_n. For arbitrary L the function J is nonincreasing, and together with the above this brackets it between two explicit values.
ER -