TEST BANK FOR Convex Optimization 1st Edition By Stephen Boyd

Let C  Rn be a convex set, with x1; : : : ; xk 2 C, and let 1; : : : ; k 2 R satisfy i  0, 1 +


+ k = 1. Show that 1x1 +


+ kxk 2 C. (The denition of convexity is that this holds for k = 2; you must show it for arbitrary k.) Hint. Use induction on k. Solution. This is readily shown by induction from the denition of convex set. We illustrate the idea for k = 3, leaving the general case to the reader. Suppose that x1; x2; x3 2 C, and 1 + 2 + 3 = 1 with 1; 2; 3  0. ...