Generalization of Schlafli formula to the volume of a spherically faced simplex

TL;DR

This paper extends the Schlafli formula to compute the volume of spherically faced simplices using identities involving Cayley-Menger determinants, derived from hypergeometric integral identities.

Contribution

It introduces new identities relating the volume of spherically faced simplices to Cayley-Menger determinants, expanding geometric volume formulas.

Findings

Derived contiguity and variation identities for simplex volume.

Expressed identities in terms of Cayley-Menger determinants.

Connected geometric volume formulas with hypergeometric integral limits.

Abstract

We present two identities (contiguity relation and variation formula) concerning the volume of a spherically faced simplex in the Euclidean space. These identities are described in terms of Cayley-Menger determinants and their differentials involved with hypersphere arrangements. They are derived as a limit of fundamental identities for hypergeometric integrals.