On a Characterisation of Inner Product Spaces

It is well known that for the Hilbert space H the minimum value of the functional F μ (f) = ∫ H ‖f – g‖2 dμ(g), f ∈ H, is achived at the mean of μ for any probability measure μ with strong second moment on H. We show that the validity of this property for measures on a normed space having support at three points with norm 1 and arbitrarily fixed positive weights implies the existence of an inner product that generates the norm.