6.30 Suppose that we are given a set of vectors {x(1),,x(p)}, x(i) n, i = 1,,p. Find the vector x n such that the average squared distance (norm) between and x(1),, x(p), is minimized. Use the SOSC to prove that the vector found above is a strict local minimizer. How is related to the centroid (or center of gravity) of the given set of points {x(1),,x(p)}? Prove the following generalization of the second-order sufficient condition: Theorem: Let be a convex subset of n, f 2 a real-valued function on , and x* a point in . Suppose that there exists c , c > 0, such that for all feasible directions d at x* (d 0), the following hold: 1. df(x*) 0. 2. dF(x*)d c||d||2. Then, x* is a strict local minimizer of f.