Non-projectability of polytope skeleta - Mathematics > Metric GeometryReport as inadecuate




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Abstract: We investigate necessary conditions for the existence of projections ofpolytopes that preserve full k-skeleta. More precisely, given the combinatoricsof a polytope and the dimension e of the target space, what are obstructions tothe existence of a geometric realization of a polytope with the givencombinatorial type such that a linear projection to e-space strictly preservesthe k-skeleton. Building on the work of Sanyal 2009, we develop a generalframework to calculate obstructions to the existence of such realizations usingtopological combinatorics. Our obstructions take the form of graph coloringsand linear integer programs. We focus on polytopes of product type andcalculate the obstructions for products of polygons, products of simplices, andwedge products of polytopes. Our results show the limitations of constructionsfor the deformed products of polygons of Sanyal and Ziegler 2009 and the wedgeproduct surfaces of R\-orig and Ziegler 2009 and complement their results.



Author: Thilo Rörig, Raman Sanyal

Source: https://arxiv.org/







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