In the present paper we study the minimization of energy integrals on the sphere with a focus on an interesting clustering phenomenon: for certain types of potentials, optimal measures are discrete or are supported on small sets. In particular, we prove that the support of any minimizer of the p-frame energy has empty interior whenever p is not an even integer. A similar effect is also demonstrated for energies with analytic potentials which are not positive definite. In addition, we establish the existence of discrete minimizers for a large class of energies, which includes energies with polynomial potentials.

中文翻译：

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球上的能量和最小化措施的离散性

在本文中，我们研究了球体上能量积分的最小化，重点是一个有趣的聚类现象：对于某些类型的势，最优测度是离散的或在小集合上得到支持。特别地，我们证明，只要p不是偶数整数，p帧能量的任何最小化器的支持内部都是空的。对于解析电位不是正定的能量，也显示出类似的效果。此外，我们建立了针对大量能量的离散最小化器的存在，其中包括具有多项式势的能量。