a) Let f : Rn R be a convex function and let c R be a fixed number. Show that the set {x Rn | f (x) c} is a convex set. Hint: use the definitions of convex functions and convex sets. (b) Show that the function f : R1 R given by f (x) = x2 is a convex function. Hint: this is an algebraic calculation. (c) Show that the function f : Rn R given by f (x) = |x|2 is a convex function. Here |x| := px2 1 + + x2 n = x x is the norm of the vector x. Hint: use the triangle inequality: |a + b| |a| + |b| for all a, b Rn and then apply part (b). (d) Use parts (a) and (c) above to show that the unit ball {x Rn | |x|2 1} in Rn is a convex set