Abstract
In this paper we investigate the extremal properties of the sum
where A i are vertices of a regular simplex, a cross-polytope (orthoplex) or a cube and M varies on a sphere concentric to the sphere circumscribed around one of the given polytopes. We give full characterization for which points on Γ the extremal values of the sum are obtained in terms of λ. In the case of the regular dodecahedron and icosahedron in \({\mathbb{R}^3}\) we obtain results for which values of λ the corresponding sum is independent of the position of M on Γ. We use elementary analytic and purely geometric methods.
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Nikolov, N., Rafailov, R. On extremums of sums of powered distances to a finite set of points. Geom Dedicata 167, 69–89 (2013). https://doi.org/10.1007/s10711-012-9804-3
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DOI: https://doi.org/10.1007/s10711-012-9804-3