Lemma is convex for . Proof Need to show WLOG . It is certainly true for (then equality holds). Now take derivative and show LHS>RHS. Minkowski’s Inequality Let and (see L norms). Then and Proof First show if then for all . For each have (using lemma) Now sum over and get that RHS is at most 1, so LHS is at most 1. Now for general , apply the inequality to and and take and get the result.