Let (X, · ) be a normed linear space over the reals. It is shown that x y x − y ≥ c_p(x^p + y^p)^1/p y x − x y (*) for all x, y X, where c_p = 2^{− 1 − 1/p} if 0 < p ≤ 1 and c_p = 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 c_p = 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.