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n Problem 4 (25 points). Let 01, ..., An E Rd. We consider the function f:Rd + R defined for x E Rd by f(x) || X 2||2 (a) Show that f is convex. (b) Show that f admits a unique global minimizer m* = +(a1 + ... + an). 1 n Remark: This proves that the mean is the vector that minimizes the average square distances to the data points. (c) We now consider the following constrained optimization problem: minimize f(x) subject to || 2 || 2 = 1. (1) We admit that (1) has a unique solution x* and we assume that m* + 0. Show that either x* = m*/||m*|| or ** = -m*/||m* ||. (d) Show that x* = m*/||m*||. Hint: express f(x) in terms of x,m* and Li ||ai||2 only