Unit Lengthenings of Tetrahedra
Abstract
In this paper we give an affirmative answer to the following question posed by Daryl Cooper: If one lengthens the sides of a tetrahedron by one unit, is the result still a tetrahedron and (if so) does the volume increase? Our proof involves a (presumably) new and sharp inequality involving the Cayley-Menger determinant and one of its directional derivatives. We give a rigorous computer-assisted proof of the inequality. We also sketch an argument which derives the existence portion of the result, in all dimensions, from an old theorem of Von Neumann. Finally, we prove a number of additional results concerning the effect on volume of selectively lengthening some of the sides of a tetrahedron.
Keywords
Cite
@article{arxiv.1407.4104,
title = {Unit Lengthenings of Tetrahedra},
author = {Richard Evan Schwartz},
journal= {arXiv preprint arXiv:1407.4104},
year = {2014}
}
Comments
36 pages, computer assisted proof. Software available from author's website. New version is an expanded version of the original, with additional results and a more canonical proof of the main result