Cutting a Simplex with a Hyperplane Question of Volume.

Simplex is a geometric figure which is extensively used in economic theory. Depending on the needs of the theory, a simplex S is sometimes cut with a hyperplane H, which separates it into two parts T and T'. It may be useful to appreciate the manner in which the simplex in cut. One mode of comparison is to evaluate the "size" of the different subsets by calculating their "volumes". The aim of this paper is to give a simple expression of the ratio : "volume" of T/"volume" of S. Furthermore, when S is the unit simplex and H goes through O, this expression becomes a nice analytical formula. By the way, we establish directly a remarkable identity of real numbers.