Power invariants of certain point sets

We consider point sets $A_1,\dots,A_n$ in the space $\mathbb R^d$, $d\ge2$, which have center of gravity at zero and, for a certain set of even exponents $2,4,\dots,2p$, “power invariants” $I_k$ in the following sense. For the sphere $S^{d-1}(R)$ with center at zero and radius $R$ and for a point $M\in S^{d-1}(R)$, the sum $I_k(M)=\sum^n_{i=1}|MA_i|^{2k}$ does not depend on the position of $M$ on the sphere $S^{d-1}(R)$ for $k=1,\dots,p$.