AI 中文总结
该研究解决了球面上给定长度简单闭曲线的平均测地距离最小化问题，给出了精确最小值及其达到条件，并确定了长度不超过2π时的下确界。
AI 中文摘要
Kimberling 问题 10 要求：在单位球面上，给定长度 $L$（特别是 $L=4\pi$）的简单闭曲线，使球面上一点到该曲线的平均测地距离 $\mathcal{J}$ 最小。对于正整数 $n$，令 $\vartheta_{n}=\pi/(2n)$，$L_{n}=2\pi/\sin\vartheta_{n}$。我们证明：在长度至多 $L_{n}$ 的可求长简单闭曲线中，$\mathcal{J}$ 的最小值等于 $\vartheta_{n}-\tan(\vartheta_{n}/2)$，且该最小值仅由长度恰好为 $L_{n}$ 的曲线取得，并且 Gerlach 和 von der Mosel 的球填充绳 $\beta^{n,k}$ 达到该最小值。Kimberling 的情形是 $n=3$：在 $L=4\pi$ 时，最小值为 $\pi/6+\sqrt{3}-2=0.255649\ldots$，由一条显式的六弧曲线及其镜像取得。对于 $L\le2\pi$，我们精确确定 $J(L)$，即长度为 $L$ 的曲线中 $\mathcal{J}$ 的下确界：它等于 $\pi/2-L/(2\pi)$，且恰好由长度为 $L$ 的圆取得。在长度 $L_{n}$ 处，我们不对所有极小曲线进行分类，但证明每一条都将球面分成两个面积为 $2\pi$、内半径为 $\vartheta_{n}$ 的圆盘，其向内邻域在每一深度都具有最大可能面积。大圆是 $n=1$ 时唯一的极小曲线，而 $\beta^{n,k}$ 在合同意义下是仅有的厚度至少为 $\sin\vartheta_{n}$ 的极小曲线。对于任意 $L$，函数 $J$ 是非增的，结合上述结果，可将它夹在两个显式值之间。
英文摘要
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 $\mathcal{J}$ from a point of the sphere to the curve. For a positive integer $n$, put $\vartheta_{n}=π/(2n)$ and $L_{n}=2π/\sin\vartheta_{n}$. We show that the minimum of $\mathcal{J}$ over rectifiable simple closed curves of length at most $L_{n}$ equals $\vartheta_{n}-\tan(\vartheta_{n}/2)$, 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 $π/6+\sqrt{3}-2=0.255649\ldots$, attained by an explicit six-arc curve and by its mirror image. For $L\le2π$ we determine $J(L)$, the infimum of $\mathcal{J}$ 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 $\vartheta_{n}$ whose inward collars have the largest possible area at every depth. The great circle is the only minimizer for $n=1$, and the $β^{n,k}$ 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.
Comments10 pages, 1 figure
