Shortest closed curve to contain a sphere in its convex hull

TL;DR

This paper proves a lower bound on the length of closed curves containing a sphere in their convex hull in 3D, generalizes a previous conjecture, and discusses higher-dimensional analogs with sharp estimates.

Contribution

It establishes a minimal length bound for such curves in 3D and extends the analysis to higher dimensions with sharp estimates, generalizing prior work.

Findings

Length of closed curve ≥ 4π in 3D

Characterization of equality case in 3D

Higher-dimensional estimate L ≥ Cn√n

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 $C$.