Skyscraper polytopes and realizations of plane triangulations

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.