The Stolarsky principle and energy optimization on the sphere. (English) Zbl 1426.11075
In the paper under review the authors study the relation between the discrepancy of a point set \(Z=\{z_1,z_2,\dots,z_N\}\subseteq \mathbb S^d\) on a sphere and the energy \[ E_F(Z)=\frac{1}{N^2}\sum_{i,j}^N F(z_i,z_j) \] for a given function \(F:\mathbb S^d\times \mathbb S^d\rightarrow \mathbb R\). In case that \(F\) is the distance function \(F(x,y)=\|x,y\|\) such a relation is known as Stolarsky invariance principle which was proved by K. B. Stolarsky [Proc. Am. Math. Soc. 41, 575–582 (1973; Zbl 0274.52012)]. In this paper, the authors prove several generalizations of the Stolarsky invariance principle and further explore the relation between the discrepancy and the energy of point sets on a sphere.
Reviewer: Volker Ziegler (Salzburg)
MSC:
| 11K38 | Irregularities of distribution, discrepancy |
| 74G65 | Energy minimization in equilibrium problems in solid mechanics |
| 42A82 | Positive definite functions in one variable harmonic analysis |
Citations:
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