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 Ee^h|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<p<2 and for any t∈E. The criterion is not correct when at least one of the conditions 0<p<2 or Ee^h|X|<∞ breaks.