On the diagonal weights of inscribed polytopes (Q376527)

The authors define the diagonal weight for a polytope as the sum of the squares of the lengths of all diagonals and sides of the polytope. They demonstrate that for a regular \(n\)-dimensional polytope with \(v\) vertices \({P_1,\dots,P_{v}}\) the diagonal weight equals \({v^2}\). Also, they show that the diagonal weight for an \(n\)-dimensional polytope with \(v\) vertices \({P_1,\dots,P_{v}}\) inscribed in a unit \(n\)-sphere and \(c\) being the distance between the centroid and the circumcenter equals \(v^2(1-c^2)\).