Majorization in Euclidean geometry and beyond

We relate the well known notion of majorization to the behavior of a pair of simplices in a Euclidean n-space. We obtain a geometrical meaning for the determinant of the involved doubly stochastic matrix. Independently, a basic theorem about volumes of simplices contained one in another, even if of different dimensions, is proved. Some related geometric questions are also presented. (C) 2014 Elsevier Inc. All rights reserved.