One characterization of symmetry

In this note we present one characterization of symmetry of probability distributions in Euclidean spaces which is formulated as follows. Let X and Y be independent and identically distributed random elements in a separable Euclidean space E. If Eeh|X|<∞, h>0, then the distribution of X is symmetric if and only if E|(X−Y,t)|p=E|(X+Y,t)|p for some 0