Diagonals of rotation matrices

AbstractAlfred Horn showed, using a theorem involving orthostochastic matrices, that the set of all diagonals of rotation matrices of order n is equal to the convex hull of those points (±1,…,±1) of which an even number (possibly 0) of coordinates are -1. He asked to prove without the use of that theorem the convexity of the set D of diagonals of rotation matrices. That is the principal goal of this work. Moreover, in the case n = 3 we study the function that associates to any rotation of order 3 its diagonal