Shortest closed curve to contain a sphere in its convex hull

We show that in Euclidean 3-space any closed curve which contains the unit sphere within its convex hull has length $L⩾4\pi$, 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 $L⩾Cn\sqrt{n}$ by Nazarov, which is sharp up to the constant $C$.