Shortest closed curve to inspect a sphere

TL;DR

This paper proves that the shortest closed curve outside a sphere, enclosing it in its convex hull, has length at least 4π, with equality only for a specific four-semicircle configuration.

Contribution

It confirms Zalgaller's 1996 conjecture by establishing the minimal length and characterizing the extremal curve for inspecting a sphere.

Findings

Shortest such curve has length at least 4π

Equality occurs only for a curve made of four semicircles

The extremal curve resembles a baseball seam shape

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.