Norm Inequalities and Characterizations of Inner Product Spaces

Let (X, || • ||) be a normed linear space over the reals. It is shown that ||x|| ||y|| ||x − y|| ≥ cp(||x||p + ||y||p)1/p || ||y|| x − ||x|| y || (∗) for all x, y ∈ X, where cp = 2− 1 − 1/p if 0 < p ≤ 1 and cp = 2−2 if p ≥ 1. The case when p = 1 is due to Dunkl and Williams. The above inequality is further investigated when the given norm is induced by an inner product. In particular, it is shown that if 0 < p ≤ 1, then (∗) holds with cp = 2 −1/p if and only if (X, || • ||) is an inner product space; this generalizes the work of Kirk and Smiley. The case, when p is infinite, is also discussed.