关于平面极小簇的 Steiner 性质。各向同性情形
On the Steiner property for planar minimizing clusters. The isotropic case
AI总结：
本文研究双密度平面簇等周问题的各向同性情形，证明极小簇在一般条件下具有 Steiner 性质，即边界由正则弧组成并在三重点以 $120^\circ$ 相交。
摘要：
我们考虑具有双密度的平面簇等周问题，即周长和体积依赖于两个权重。本文研究各向同性情形，平行论文《On the Steiner property for planar minimizing clusters. The anisotropic case》则研究各向异性情形。我们证明，在相当一般的条件下，极小簇具有“Steiner 性质”，即其边界由 ${\rm C}^{1,\gamma}$ 正则弧组成，这些弧在有限多个三重点交汇，且满足 $120^\circ$ 性质。
英文摘要：
We consider the isoperimetric problem for clusters in the plane with a double density, that is, perimeter and volume depend on two weights. In this paper we consider the isotropic case, in the parallel paper "On the Steiner property for planar minimizing clusters. The anisotropic case", the anisotropic case is studied. Here we prove that, in a wide generality, minimal clusters enjoy the "Steiner property", which means that the boundaries are made by ${\rm C}^{1,γ}$ regular arcs, meeting in finitely many triple points with the $120^\circ$ property.
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