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Consider a set $A = \{ a_1,a_2,\ldots, a_n \} $ of vectors in $\mathbb{R}^d$, which lie in a common affine hyperplane. Two convex polytopes may be obtained from $A$, namely the convex hull of the vectors in $A$, $conv(A)$ and the zonotope generated by vectors in $A$, $Z(A)$. Both polytopes can be viewed as the projection of a polytope $P$ which sends the unit vectors in $\mathbb{R}^n$ to the columns of $A$, where $P$ is the standard $n-1$-simplex in the first case and the unit $n$-cube in the second. The general study of affine projections of convex polytopes is developed in this paper of Billera and Sturmfels entitled "Fiber Polytopes"

The first question may be common knowledge, but is there some relation between the triangulations of $conv(A)$ and cubical tilings of $Z(A)$ which arise from this projection? I haven't seen this explicitly written down, but in the light of fiber polytopes both should be related concepts.

Question: Is there a reason why the triangulations of $conv(A)$ should be considered in some sense "more natural" than the cubical tilings of $Z(A)$ (both subdivisions seen as arising from the projection)?.

The reason I ask this is because I have seen many instances in the literature of toric ideals where triangulations of a convex polytope are used to characterize some algebraic construction but, as far as I remember, none performing similar characterizations in terms of tilings of zonotopes. As an example, a theorem of Sturmfels here characterizes the radicals of the initial ideals of a toric variety (associated to a matrix $A\in \mathbb{Z}^{d\times n}$) as the radical of the Stanley-Reisner ideals of regular triangulations of the convex hull of the columns of $A$.

However, when it comes to the combinatorial information of a vector configuration, the zonotope associated to it seems to relate more directly to the oriented matroid of the vector configuration. Recall, for instance the Bohne-Dress theorem relating the set of zonotopal tilings of the zonotope generated by the vector configuration and the one-element liftings of its oriented matroid.

I would be very satisfied with answers which, say,

  • give pointers to characterization of algebraic objects (e.g. Groebner bases, minimal free resolutions) from the theory of toric ideals in terms of some associated zonotopes.
  • give pointers to a general relation between triangulations of $conv(A)$ and cubical tilings of $Z(A)$,
  • indicate why, and in which context, one of both concepts should be more natural than the other.

p.s. may be someone reputed enough would like to create the tag oriented-matroids?

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Maybe the connection equivalence of unimodular triangulations and square-free initial ideals is already the best argument why triangulations are 'better'. – Thomas Kahle Apr 11 '12 at 15:53
For me this would rather be a disadvantage, since it means that triangulations can only be associated with square-free initial ideals (in the general case, one would have to go to the radical to get a triangulation). – Camilo Sarmiento Apr 12 '12 at 8:51
up vote 3 down vote accepted

Triangulations of polytopes are "more fundamental" than cubical tilings of zonotopes. By the Cayley trick, every cubical tiling of a zonotope can be seen as a triangulation of the Cayley lifting of the segments defining it. The latter is equal to their Lawrence lifting.

This can be seen as an addition to the Bohne-Dress theorem, relating tilings of a zonotope to liftings of the associated oriented matroid.

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