CHARACTERIZATIONS OF INNER PRODUCT SPACES BY STRONGLY CONVEX FUNCTIONS

New characterizations of inner product spaces among normed spaces involving the notion of strong convexity are given. In particular, it is shown that the following conditions are equivalent: (1) (X,k · k) is an inner product space; (2) f : X ! R is strongly convex with modulus c > 0 if and only if f ck · k 2 is convex; (3) k · k 2 is strongly convex with modulus 1.