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 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 4 n 3 × 8 n 5 × ζ ( n ) integer grid, where ζ ( n ) ≤ ( 500 n 8 ) τ ( G ) and τ ( G ) denotes the shedding diameter of G , a quantity defined in the paper.

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