The Stolarsky Principle and Energy Optimization on the Sphere
The Stolarsky Principle and Energy Optimization on the Sphere
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DOI：
10.1007/s00365-017-9412-4
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发表时间：
2018-08-01
影响因子：
2.7
通讯作者：
Matzke, Ryan
中科院分区：
文献类型：
--
作者：
Bilyk, Dmitriy;Dai, Feng;Matzke, Ryan
经典的Stolarsky不变性原理将球体上有限点集的球冠差异与点之间的欧几里得距离的成对和联系起来。在本文中，我们进一步探讨和扩展这一现象。除了这一事实的一个新的基本证明，我们建立了几个新的类似物，其中涉及到不同的离散能量的差异的各种概念。特别地，我们发现半球差异与测地距离之和有关。我们还将这些结果扩展到任意措施的领域和任意概念的差异，并将它们应用到问题的能源优化和组合几何，并发现，令人惊讶的是，测地距离能源的行为不同于其欧几里德对应。
The classical Stolarsky invariance principle connects the spherical cap discrepancy of a finite point set on the sphere to the pairwise sum of Euclidean distances between the points. In this paper, we further explore and extend this phenomenon. In addition to a new elementary proof of this fact, we establish several new analogs, which relate various notions of discrepancy to different discrete energies. In particular, we find that the hemisphere discrepancy is related to the sum of geodesic distances. We also extend these results to arbitrary measures on the sphere and arbitrary notions of discrepancy and apply them to problems of energy optimization and combinatorial geometry and find that, surprisingly, the geodesic distance energy behaves differently than its Euclidean counterpart.