The sine theorem and inequalities for volumes of simplices and determinants (Q1890751)

The author provides two simple proofs of the sine theorem for \(n\)- dimensional simplices, which is used, together with an inequality of L. Fejes Tóth, to prove a volume majorization for an \(n\)-simplex in terms of the \((n-1)\)-dimensional content of its faces. Several corollaries are proved, one of them providing a volume majorization for an \(n\)-simplex in terms of the distances between its vertices. This corollary can be regarded as an improvement of the Hadamard (1893) inequality. Given that most of these results are known, the main merit of the present paper lies in the presentation of self-contained elementary proofs.