An analytical approach to the Rational Simplex Problem

Abstract

In 1973, J. Cheeger and J. Simons raised the following question that still remains open and is known as the Rational Simplex Problem: Given a geodesic simplex in the spherical 3-space so that all of its interior dihedral angles are rational multiples of $\pi$, is it true that its volume is a rational multiple of the volume of the 3-sphere? We propose an analytical approach to the Rational Simplex Problem by deriving a function $f(t)$, defined as an integral of an elementary function, such that if there is a rational $t$, close enough to zero, such that the value $f(t)$ is an irrational number then the answer to the Rational Simplex Problem is negative.

5 pages; in version 3, several new references are added and discussed

References in corpus (4)

• Lower bounds on volumes of hyperbolic Haken 3-manifolds
• The volume formulas for a spherical tetrahedron
• Spherical simplices generating discrete reflection groups
• An elementary proof of the irrationality of Tschakaloff series