The intrinsic diameter of the surface of a parallelepiped (Q1006389)

In Euclidean \(3\)-space, a convex surface is the boundary of a compact convex body. The intrinsic metric between two points of such a surface is the length of the shortest path between them and the intrinsic diameter is the maximal intrinsic distance. The authors give explicit formulas for the intrinsic diameter of an arbitrary rectangular parallelepiped. There are three cases depending on the relative sizes of the edge lengths \(a,b,c\). The relationships distinguishing these cases and the consequent formulas for the diameter are extremely complicated (\textit{e.g.} one is the solution of a polynomial of degree eight in \(a,b,c\)). A corollary is that among all parallelepipeds with given intrinsic diameter the one with maximal surface area has \(a:b:c = 1:1:\sqrt{2}\).