vectors in Rm (a) Let (Carathodory's theorem) Let A,..., An be a collection of -{ i=1...

  

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vectors in Rm (a) Let (Carathéodory's theorem) Let A₁,..., 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 A₁,..., An ≥ | 20}. - P = n (A₁,..., An) i Ai = y, A₁,..., An ≥ ~ (b) Let P be the convex hull of the vectors A₁: n {E₁ ΣAi Ai Ai ΕΧΕΙ i=1 i=1 n i=1 ... >0}. +20}. 1, A1,..., An > ܕ 1 AiAi, where Show that any element of P can be expressed in the form Σ₁₁ di = = 1 and ₂ ≥ 0 for all i, with at most m + 1 of the coefficients Xi being nonzero. vectors in Rm (a) Let (Carathéodory's theorem) Let A₁,..., 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 A₁,..., An ≥ | 20}. - P = n (A₁,..., An) i Ai = y, A₁,..., An ≥ ~ (b) Let P be the convex hull of the vectors A₁: n {E₁ ΣAi Ai Ai ΕΧΕΙ i=1 i=1 n i=1 ... >0}. +20}. 1, A1,..., An > ܕ 1 AiAi, where Show that any element of P can be expressed in the form Σ₁₁ di = = 1 and ₂ ≥ 0 for all i, with at most m + 1 of the coefficients Xi being nonzero.