An open problem on metric invariants of tetrahedra

Abstract

In ISSAC 2000, P. Lisoněk and R.B. Israel [3] asked whether, for any given positive real constants V,R,A1,A2,A3,A4, there are always finitely many tetrahedra, all having these values as their respective volume, circumradius and four face areas. In this paper we present a negative solution to this problem by constructing a family of tetrahedra T(x,y) where $(x,y)$ varies over a component of a cubic curve such that all tetrahedra T(x,y) share the same volume, circumradius and face areas.