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Zonotopal tilings and the Bohne-Dress theorem
We prove a natural bijection between the polytopal tilings of a zonotope Z by zonotopes, and the one-element-liftings of the oriented matroid M(Z) associated with Z. This yields a simple proof and a strengthening of the Bohne-Dress Theorem on zonotopal tilings. Furthermore we prove that not every oriented matroid can be represented by a zonotopal tiling. Introduction. At the 1989 Stockholm "Symposium on Combinatorics and Geometry", Andreas Dress announced the following surprising theorem: Thedoi:10.1090/conm/178/01902 fatcat:y2zdaiua2vanrbwyiau2nb2tzm