论文标题
飞机中点的平衡配置
Balanced configurations of points in the plane
论文作者
论文摘要
$ s^2 $上点的平衡配置是一组（有限的）点，如果仅根据两个点之间的距离，它们相互作用，则它们处于平衡状态。如果在配置中的每个点$ \ Mathcal {C} $中的每个点，则配置是组合的，则只有$ \ MATHCAL {C} $仅修复了该点及其对立的对称性。 Leech表明，通过对所有可能的平衡配置进行分类，这些定义在Sphere $ S^2 $上等效。另一方面，Cohn，Elkies，Kumar和Schürmann表明，对于$ n \ geq 7，$ $ s^{n-1} $中有平衡配置的示例，这些配置不是均不是平衡的。他们还建议将平衡配置的概念扩展到欧几里得空间，并推测至少在平面的情况下，所有离散平衡的配置均以$ \ mathbb {r}^n $为组平衡。我们通过在$ \ mathbb {r}^2 $中提供平衡配置的完整分类来验证该猜想的重新重新分类。
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.
