Zonotopal Subdivisions of Cyclic Zonotopes

The cyclic zonotope …n; d† is the zonotope in R d generated by any n distinct vectors of the form …1; t; t 2 ;. .. ; t dÀ1 †. It is proved that the re¢nement poset of all proper zonotopal subdivisions of …n; d† which are induced by the canonical projection p: …n; d H † 3 …n; d†, in the sense of Billera and Sturmfels, is homotopy equivalent to a sphere and that any zonotopal subdivision of …n; d† is shellable. The ¢rst statement gives an af¢rmative answer to the generalized Baues problem in a new special case and re¢nes a theorem of Sturmfels and Ziegler on the extension space of an alternating oriented matroid. An important ingredient in the proofs is the fact that all zonotopal subdivisions of …n; d† are stackable in a suitable direction. It is shown that, in general, a zonotopal subdivision is stackable in a given direction if and only if a certain associated oriented matroid program is Euclidean, in the sense of Edmonds and Mandel.