Zonotopal tilings and the Bohne-Dress theorem [unknown]

Jürgen Richter-Gebert, Günter M. Ziegler
1994 Contemporary Mathematics   unpublished
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: The
more » ... sing theorem: The tilings of a zonotope Z by zonotopes are in bijection with the single-element liftings of the associated oriented matroid L(Z) [5]. A proof was provided in the 1992 doctoral dissertation of Jochen Bohne [3]. (A zonotope is, to give three equivalent characterizations, a projection of a cube, a Minkowski sum of line segments, and a polytope all of whose faces are centrally symmetric [4] [14] [2, Sect. 2.2].) 1991 Mathematics Subject Classification. 05B35, 52B40; Secondary 52C22, 52B11.
doi:10.1090/conm/178/01902 fatcat:y2zdaiua2vanrbwyiau2nb2tzm