The length of the cut locus on convex surfaces (Q6568715)

In [Adv. Geom. 5, No. 1, 97--106 (2005; Zbl 1077.53055)], \textit{J.-i. Itoh} and \textit{T. Zamfirescu} conjectured that for any smooth convex surface \(S\), if \(M\) is a finite subset of \(S\) with at least two points, then the length of the cut locus of \(M\) is at least half the diameter of the surface \(S\). This paper proves this conjecture for an arbitrary convex surface. There are examples of non-convex surfaces for which the conjecture fails.