Shortest closed curve to inspect a sphere

Abstract

We show that in Euclidean 3-space any closed curve which lies outside the unit sphere and contains the sphere within its convex hull has length at least $4\pi$. Equality holds only when the curve is composed of $4$ semicircles of length $\pi$, arranged in the shape of a baseball seam, as conjectured by V. A. Zalgaller in 1996.

25 pages, 9 figures; The proof has been simplified and shortened; Accepted for publication in J. Reine Angew. Math. (Crelle's Journal)