G. Chelidze

On a Characterisation of Inner Product Spaces

\par It is well known that for the Hilbert space $H$ the minimum value of the functional $F_\mu(f)=\int_H\|f-g\|^2d\mu(g),$ $f\in H,$ is achived at the mean of $\mu$ for any probability measure $\mu$ with strong second moment on $H.$ We show that the vrlidity 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.