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A Quantitative Steinitz Theorem for Plane Triangulations | | Igor Pak
; Stedman Wilson
; | | Date: |
4 Nov 2013 | | 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 $mathbb{R}^2$ 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 $4n^3 imes 8n^5 imes zeta(n)$ integer grid, where $zeta(n)
leq (500 n^8)^{ au(G)}$ and $ au(G)$ denotes the shedding diameter of $G$, a
quantity defined in the paper. | | Source: | arXiv, 1311.0558 | | Services: | Forum | Review | PDF | Favorites | |
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