Skyscraper polytopes and realizations of plane triangulations

Abstract

We give a new proof of Steinitz's classical theorem in the case of plane triangulations, which allows us to obtain a new general bound on the grid size of the simplicial polytope realizing a given triangulation, subexponential in a number of special cases.

Formally, we prove that every plane triangulation G with n vertices can be embedded in ${\mathbb{R}}^2$ in such a way that it is the vertical projection of a convex polyhedral surface. We show that the vertices of this surface may be placed in a $4n^3\times 8n^5\times \zeta (n)$ integer grid, where $\zeta (n)\le {(500n^8)}^{\tau (G)}$ and $\tau (G)$ denotes the shedding diameter of G, a quantity defined in the paper.