凸多面体上的割轨迹实现
Cut Locus Realizations on Convex Polyhedra
摘要：
我们证明了每一个正权树T都可以实现为凸多面体P上点x的割轨迹C（x），其中T个权匹配C（x）的长度。如果T有n个叶，则P有n+1个顶点。我们表明，实际上有一个连续的多面体P每个实现T的一些X上的P。三个主要的工具在证明中的性质的明星展开的P，亚历山德罗夫的胶合定理，和切割轨迹分区引理。从T构造P是非常简单的。
英文摘要：
We prove that every positively-weighted tree T can be realized as the cut locus C(x) of a point x on a convex polyhedron P, with T weights matching C(x) lengths. If T has n leaves, P has (in general) n+1 vertices. We show there are in fact a continuum of polyhedra P each realizing T for some x on P. Three main tools in the proof are properties of the star unfolding of P, Alexandrov's gluing theorem, and a cut-locus partition lemma. The construction of P from T is surprisingly simple.
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