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arXiv · 2209.05988

Shortest closed curve to contain a sphere in its convex hull

Mohammad Ghomi·James Wenk
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Abstract

We show that in Euclidean 3-space any closed curve $\gamma$ which contains the unit sphere within its convex hull has length $L\geq4\pi$, 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 $L\geq Cn\sqrt{n}$ by Nazarov, which is sharp up to the constant $C$.

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Mohammad Ghomi, James Wenk. 2022-09-13. Shortest closed curve to contain a sphere in its convex hull. https://arxiv.org/abs/2209.05988

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