Abstract

The cyclic zonotope Z…n; d † is the zonotope in R d generated by any ndistinct vectors of the form …1; t; t2;...; td 1 †. It is proved that the refinement poset of all proper zonotopal subdivisions of Z…n; d † which are induced by the canonical projection p: Z…n; d0 †!Z…n; d†, in the sense of Billera and Sturmfels, is homotopy equivalent to a sphere and that any zonotopal subdivision of Z…n; d † is shellable. The first statement gives an affirmative answer to the generalized Baues problem in a new special case and refines 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 Z…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.

Citations