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](https://www.jstor.org/stable/10.2307/2946575) 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.



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> **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)?.
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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](https://projecteuclid.org/journals/tohoku-mathematical-journal/volume-43/issue-2/Gr%C3%B6bner-bases-of-toric-varieties/10.2748/tmj/1178227496.full) *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?



Triangulations of polytopes and tilings of zonotopes
Triangulations of polytopes are "more fundamental" than cubical tilings of zonotopes. By the [Cayley trick](http://www.ams.org/mathscinet-getitem?mr=1763304), 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](http://www.ams.org/mathscinet-getitem?mr=1310586), relating tilings of a zonotope to liftings of the associated oriented matroid.


