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Power invariants of certain point sets. (English. Russian original) Zbl 1003.51008
J. Math. Sci., New York 110, No. 4, 2755-2768 (2002); translation from Zap. Nauchn. Semin. POMI 261, 7-30 (1999).
Summary: Point sets $$\{A_1,\dots, A_n\}$$ in $$\mathbb{R}^d$$, $$d\geq 2$$, are considered that have barycenter at the origin and, for a certain collection of even exponents $$2,4,\dots, 2p$$, possess “power invariants” $$I_k$$ in the following sense. Let $$S^{d-1}(R)$$ be the sphere with center at the origin and radius $$R$$ and let $$M\in S^{d-1}(R)$$. Then the sums $$I_k(R)= \sum^n_{i=1}|MA_i |^{2k}$$, $$k=1, \dots,p$$, do not depend on the position of $$M$$ on $$S^{d-1}(R)$$.

##### MSC:
 51M04 Elementary problems in Euclidean geometries 52A99 General convexity
##### Keywords:
point sets in $$\mathbb{R}^d$$; power invariants