GEOMETRIC CONSTANTS AND CHARACTERIZATIONS OF INNER PRODUCT SPACES

Let X be a real normed space, let Psi(2) denote the set of all convex functions on [0,1] such that max{1 - t,t} <= psi(t) <= 1, and let Phi(2) denote the set of all concave function on [0,1] such that psi(0) = psi(1) = 1. For each psi is an element of Phi(2) boolean OR Psi(2), it is shown that parallel to parallel to x parallel to(-1)x+parallel to y parallel to(-1)y parallel to <= C-psi parallel to x-y parallel to parallel to(x,y)parallel to(-1)(psi) for all nonzero vectors x, y is an element of X, where C-psi = 4 max psi(t). The case of psi = psi(p) (p > 0), defined as psi(p)(t) = ((1 - t)(p) + t(p))(1/p), is due to Al-Rashed, and is due to Dunkl and Williams when p = 1. In particular, it is shown that for certain psi is an element of Phi(2), the inequality holds for C-psi = 2 psi(1/2) if and only if X is an inner product space; this generalizes the works of Al-Rashed and Kirk-Smiley.