  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 $C$.
