On extremums of sums of powered distances to a finite set of points

Abstract

In this paper we investigate the extremal properties of the sumwhere 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 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.2000 Mathematics Subject Classification. Primary: 52A40.

Asymptotics of Discrete Riesz d-Polarization on Subsets of d-Dimensional Manifolds

“…In view of (19) relation (20) also holds when D ⊂ A is closed and H d (D) = 0. Then in view of Remark 2.1 we have (15).…”

Constant Riesz potentials on a circle in a plane with an application to polarization optimality problems

“…. , x N } in 2 and a given constant s ∈ , we define the Riesz potential function d . In this paper, we consider two problems concerning the Riesz s-potential functions U s (·; ω N ).…”

Polarization problem on a higher-dimensional sphere for a simplex

“…We remark that part of the case d = 2 of Theorem 2.1 follows from the results in [19,1,14,2,7,12] and the case d = 3 for potential functions of form (4) follows by combining the result of [21] (which is the case d = 3 of Lemmas 5.3 and 5.4) with one of the results from [15]. The case d = 3 for general potentials follows by combining the result from [21] with the assertion of Theorem 2.4 below.…”

“…3 ) is achieved at points of the set −ω * 3 (antipodes of the points from ω * 3 ), then ω * 3 is optimal for the maximal polarization problem for N = 4 points on S 2 . However, up to this point, the absolute minimum of p f (x, ω * d ) was shown to be achieved at points of −ω * d only for f of form (4), see [15] and references therein. Furthermore, one can construct potential functions f non-increasing and convex on (0, 4] such that the potential p f (x, ω * d ), d ≥ 2, does not achieve its absolute minimum over S d−1 at points of −ω * d (see Corollary 2.5 below).…”

“…with s > −2, s = 0, see the thesis by Su [21] and the result of Nikolov and Rafailov [15] finding quantity (2) for a regular simplex and potential functions of form (4). The optimality of the configuration of the vertices of a regular simplex inscribed in S 2 was proved.…”