The generalized sine theorem and inequalities for simplices (Q1307292)

The authors establish a generalized sine theorem for the Euclidean \(n\)-simplex \(S\), i.e., they extend the known \(n\)-dimensional sine theorem (given in terms of the volume, the facet areas and vertex angles of \(S)\) to \(k\)-order vertex angles (e.g., for \(k=2\) yielding a corresponding theorem about dihedral angles). They apply this generalized theorem by deriving some inequalities in terms of the \(k\)-order vertex angles of \(S\), these inequalities also yielding several characterizations of regular \(n\)-simplices. For proving these results, they use informations about the inverse matrix of the metric matrix of an \(n\)-parallelotope.