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arXiv · 2102.11097

Cut Locus Realizations on Convex Polyhedra

Joseph O'Rourke·Costin Vîlcu
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Abstract

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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Joseph O'Rourke, Costin Vîlcu. 2021-02-22. Cut Locus Realizations on Convex Polyhedra. https://arxiv.org/abs/2102.11097

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