Non-projectability of polytope skeleta

Details Non-projectability of polytope skeleta Non-projectability of polytope skeleta

2009-08-06

arxiv.org/abs/0908.0845v1

Mathematics

Metric Geometry

18 pages, 2 figures

Scientific paper

We investigate necessary conditions for the existence of projections of polytopes that preserve full k-skeleta. More precisely, given the combinatorics of a polytope and the dimension e of the target space, what are obstructions to the existence of a geometric realization of a polytope with the given combinatorial type such that a linear projection to e-space strictly preserves the k-skeleton. Building on the work of Sanyal (2009), we develop a general framework to calculate obstructions to the existence of such realizations using topological combinatorics. Our obstructions take the form of graph colorings and linear integer programs. We focus on polytopes of product type and calculate the obstructions for products of polygons, products of simplices, and wedge products of polytopes. Our results show the limitations of constructions for the deformed products of polygons of Sanyal & Ziegler (2009) and the wedge product surfaces of R\"orig & Ziegler (2009) and complement their results.

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