Let S be the simplex generated by the finite set of points E = (x1; x2; . . . ; xn). Show that each of the vertices xi is an extreme point of the simplex.
Simplexes are the most elementary of convex sets and every convex set is the union of simplexes. For this reason results are often established for simplexes and then extended to more general sets (example 1.100, exercise 1.229). The dimension of a simplex with n vertices is n - 1. Since the vertices of a simplex are a½nely independent, each element in a simplex has a unique representation as a convex combination of the vertices (exercise 1.159). The coe½cients in this representation are called the barycentric coordinates of the point.