Skyscraper polytopes and realizations of plane triangulations

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 R² 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 4n³×8n⁵×ζ(n) integer grid, where ζ(n)≤(500n⁸)^τ(G) and τ(G) denotes the shedding diameter of G, a quantity defined in the paper.