平面中点的平衡构形
Balanced configurations of points in the plane
摘要：
球面$S^2$上点的平衡构形是一组（有限的）点，如果它们根据任何只取决于两点之间距离的力定律相互作用，则这些点处于平衡状态。如果对于配置$\mathcal{C}$中的每个点，存在仅固定该点及其对极的$\mathcal{C}$的对称性，则该配置另外是组平衡的。Leech通过对所有可能的平衡构型进行分类，证明了这些定义在球面$S^2$上是等价的。另一方面，Cohn、Elkies、Kumar和Schürmann指出，对于$n\geq 7,$，在$S^{n-1}$中有平衡构型的例子，但它们不是群平衡的。他们还建议将平衡构形的概念扩展到欧几里得空间，并证明至少在平面的情况下，$\mathbb{R}^n$中所有离散的平衡构形都是群平衡的。我们验证了这一猜想的重新提供一个完整的分类的平衡配置在$\mathbb{R}^2$满足一定的最小距离属性。
英文摘要：
A balanced configuration of points on the sphere $S^2$ is a (finite) set of points which are in equilibrium if they act on each other according any force law dependent only on the distance between two points. The configuration is additionally group-balanced if for each point in a configuration $\mathcal{C}$, there is a symmetry of $\mathcal{C}$ fixing only that point and its antipode. Leech showed that these definitions are equivalent on the sphere $S^2$ by classifying all possible balanced configurations. On the other hand, Cohn, Elkies, Kumar, and Schürmann showed that for $n\geq 7,$ there are examples of balanced configurations in $S^{n-1}$ which are not group balanced. They also suggested extending the notion of balanced configurations to Euclidean space, and conjectured that at least in the case of the plane, all discrete balanced configurations in $\mathbb{R}^n$ are group-balanced. We verify a reformulation of this conjecture by providing a complete classification of the balanced configurations in $\mathbb{R}^2$ satisfying a certain minimal distance property.
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