Cut locus realizations on convex polyhedra

We prove that every positively weighted tree T can be realized as the cut locus $C\left(x\right)$ of a point x on a convex polyhedron P, with T edge weights matching $C\left(x\right)$ edge lengths. If T has n leaves, P has (in general) $n+1$ vertices. We show there is in fact a continuum of polyhedra P each realizing T for some $x\in P$. Three main tools in the proof are properties of the star unfolding of P, Alexandrov's gluing theorem, and a new cut-locus partition lemma. The construction of P from T is surprisingly simple.