Unit Lengthenings of Tetrahedra

TL;DR

This paper proves that increasing all sides of a tetrahedron by one unit results in a larger volume, using a new inequality involving the Cayley-Menger determinant and computer-assisted proof techniques.

Contribution

It provides a new inequality involving the Cayley-Menger determinant and its derivatives, and confirms that lengthening all sides of a tetrahedron increases its volume.

Findings

Lengthening all sides of a tetrahedron increases its volume.

A new sharp inequality involving the Cayley-Menger determinant was established.

Computer-assisted proof techniques were employed to verify the inequality.

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.