A Quantitative Steinitz Theorem for Plane Triangulations

2013/11/04 by Igor Pak, Pak, Igor, Stedman Wilson +1
Computer Science · Mathematics · #05C62 (Primary) #52B10 #68R10 (Secondary) #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #Computational Geometry (cs.CG) #Computational Geometry and Mesh Generation #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1311.0558

openalex publication_date 2013/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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) ≤ (500 n⁸)^τ(G) and τ(G) denotes the shedding diameter of G, a quantity defined in the paper.

Related

• A better upper bound on the number of triangulations of a planar point set
• The domination number of plane triangulations
• Regular flip equivalence of surface triangulations
• A lower bound technique for triangulations of simplotopes
• A subexponential size triangulation of ℝPⁿ