An inequality for \(n\)-dimensional sines of vertex angles of a simplex with some applications (Q1904270)

The \(n\)-dimensional sine \(S_i\) of the \(i\)-th vertex angle of a simplex, defined in 1978 by Eriksson, is the dimensionless value \((n!V)^{n - 1} \prod [(n - 1)! V_j]^{- 1}\), where \(V\) is the \(n\)-dimensional measure of the simplex, \(V_j\) is the \((n - 1)\)-dimensional measure of a face of the simplex containing the \(i\)-th vertex, and the product runs over all such faces. It is easily seen that in the case \(n = 2\) this reduces to the sine of the angle. The authors prove a very flexible inequality involving the \(n\)-dimensional sines of an arbitrary simplex and an arbitrary \((n + 1)\)-tuple of real numbers, using techniques from linear algebra and the AGM inequality. They then prove several old and new geometric inequalities as simple corollaries of the main theorem. A particularly amusing result: the least upper bound (over all simplexes in all dimensions) of \(\Sigma S_i^2\) is \(e\).