Cut Locus Realizations on Convex Polyhedra

TL;DR

This paper demonstrates that any positively-weighted tree can be realized as the cut locus of a point on a convex polyhedron, providing a constructive method and showing the existence of infinitely many such polyhedra.

Contribution

It establishes a universal realization result for weighted trees as cut loci on convex polyhedra, expanding understanding of geometric structures and their properties.

Findings

Any positively-weighted tree can be realized as a cut locus on some convex polyhedron.

There are infinitely many convex polyhedra realizing the same tree as a cut locus.

The construction method is simple and relies on properties of star unfoldings, Alexandrov's theorem, and a cut-locus partition lemma.

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.