The Intrinsic Diameter of the Surface of a Parallelepiped

Abstract

In the paper we obtain an explicit formula for the intrinsic diameter of the surface of a rectangular parallelepiped in 3-dimensional Euclidean space. As a consequence, we prove that an parallelepiped with relation 1 : 1 : √ 2 for its edge lengths has maximal surface area among all rectangular parallelepipeds with given intrinsic diameter.

Farthest points on flat surfaces

“…Note that this is the only possible case if b ≤ 2a. From (5), Figure 3; φ (c + ) and φ (c − ) are the only points of ramification of the cut-locus of p. A simple computation shows that…”

“…The survey [13], though a little dated, is still a reference in the field. It should be noted that few examples of explicit computations of farthest points are known, see [6] or even [5], and the counterexamples to the mentioned conjecture of Steinhauss [12], [14], [3], [4]. On the other hand, no convex polyhedron has F a single-valued involution [7].…”

Antipodal Points and Diameter of a Sphere

“…This theorem is helpful in investigating diameters for closed surfaces in R 3 with central symmetry. In particular, Theorem 1 has applications in computational geometry for evaluation of diameter for a surface of a rectangular parallelepiped [4].…”

Alexandrov's isodiametric conjecture and the cut locus of a surface

“…Theorem 1 follows as a weaker version of this theorem by neglecting the non-positive term in the brackets and applying the crude bound (15).…”

“…Even within restricted families of surfaces, such as the boundary of parallelepipeds, the problem does not seem to be easy, mainly due to the difficulties arising in the computation of the intrinsic diameter of a given surface. The classes for which the conjecture has been proved are tetrahedra (Makai [13] and Zalgaller [22]), rectangular parallelepipeds (Nikorov and Nikorova [15]) and surfaces of revolution (Makuha [14] and Abreu and the first author [1]). Except for the last family of surfaces, which contains the double disk, the optimizers in the first and second cases are not degenerate, being the regular tetrahedron and the parallelepiped with edge lengths 1, 1 and √ 2, respectively, with the corresponding inequalities satisfied for surfaces within each family being…”