In this note, we present certain characterizations of inner product
spaces which are based on geometric norm equalities. In particular, we show that a
normed linear space X is an inner product space if and only if for each x, y 2 X with
kxk = kyk = 1, there exists t 2 (0, 1/2) such that k(1 ! t)x + tyk = ktx + (1 ! t)yk
holds.