Vertex angles for simplices (Q1809004)

Let \(\mathbb E^n\) be the \(n\)-dimensional Euclidean space with an orientation. Let \(\Omega\) be an \(n\)-simplex in \(\mathbb E^n\) and let \(\Omega_i\) be its facet which lies in a hyperplane \(\Pi_i\), \(e_i\) the unit outer normal vector of \(\Pi_i\), \(D_i = \det (e_0,\dots,e_{i-1}, e_{i+1},\dots, e_n)\), \(i= 0,1,\dots,n.\) The angle \(\theta_i = \arcsin |D_i|\) is called the vertex angle at vertex \(A_i\) of \(\Omega.\) It is well known that a simplex is regular iff all its dihedral angles are equal. A natural question is whether a similar result holds for vertex angles. The authors obtain the following solution to this problem. For any regular \(n\)-simplex, \[ \theta_i = \arcsin[(1 + 1/n)^n/(n + 1)]^{1/2}, \quad i= 0,1,\dots,n; \] there exists a nonregular tetrahedron \(\Omega^\ast\) in \(\mathbb E^3\) such that all vertex angles of \(\Omega^\ast\) are equal. The authors also prove that for a rectangular \(n\)-simplex, \(\sum_{i=0}^n\sin^2\theta_i=2\) holds. Let \(\{x_1,x_2,\dots,x_n\}\) be a set of orthogonal nonzero vectors in \(\mathbb R^n,\) \(U\) the \(k\)-dimensional subspace of \(\mathbb R^n.\) Then \(\sum_{i=0}^n\sin^2(x_i,U)=n-k.\)