A QUANTITATIVE STEINITZ THEOREM FOR 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)(G) and τ(G) denotes the shedding diameter of G, a quantity defined in the paper