vectors in Rm (a) Let (Carathodory's theorem) Let A,..., An be a collection of -{ i=1 A = C = Show that any element of C can be expressed in the form 1 AiAi, with Ai 0, and with at most m of the coefficients A, being nonzero. Hint: Consider the polyhedron = {...... Ai Ai A,..., An | 20}. - P = n (A,..., An) i Ai = y, A,..., An ~ (b) Let P be the convex hull of the vectors A: n {E Ai Ai Ai i=1 i=1 n i=1 ... >0}. +20}. 1, A1,..., An > 1 AiAi, where Show that any element of P can be expressed in the form di = = 1 and 0 for all i, with at most m + 1 of the coefficients Xi being nonzero.