Shortest closed curve to contain a sphere in its convex hull

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2024-05-13 DOI:10.1112/blms.13066

Mohammad Ghomi, James Wenk

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Abstract

We show that in Euclidean 3-space any closed curve which contains the unit sphere within its convex hull has length $L ⩾ 4 π$ , and characterize the case of equality. This result generalizes the authors' recent solution to a conjecture of Zalgaller. Furthermore, for the analogous problem in $n$ dimensions, we include the estimate $L ⩾ C n \sqrt{n}$ by Nazarov, which is sharp up to the constant $C$ .

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将球面包含在其凸面内的最短闭合曲线

我们证明了在欧几里得三维空间中，任何在其凸壳内包含单位球面的闭合曲线都有长度 L ⩾ 4 π $L\geqslant 4\pi$ ，并描述了相等情况的特征。这一结果概括了作者最近对扎尔加勒猜想的解答。此外，对于 n $n$ 维度的类似问题，我们包含了纳扎罗夫的估计 L ⩾ C n n $L\geqslant Cn\sqrt {n}$，它在常数 C $C$ 的范围内是尖锐的。

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