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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 \(S^{d-1}(R)\).

51M04 Elementary problems in Euclidean geometries
52A99 General convexity