Triangulations of polytopes and tilings of zonotopes - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T22:49:31Z http://mathoverflow.net/feeds/question/93772 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/93772/triangulations-of-polytopes-and-tilings-of-zonotopes Triangulations of polytopes and tilings of zonotopes Camilo Sarmiento 2012-04-11T14:16:24Z 2013-02-21T08:06:15Z <p>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 <a href="http://www.jstor.org/stable/10.2307/2946575" rel="nofollow">paper</a> of Billera and Sturmfels entitled "Fiber Polytopes" </p> <p>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.</p> <blockquote> <p><strong>Question:</strong> 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)?.</p> </blockquote> <p>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 <a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.tmj/1178227496" rel="nofollow">here</a> <em>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 $A$</em>.</p> <p>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.</p> <p>I would be very satisfied with answers which, say,</p> <ul> <li>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.</li> <li>give pointers to a general relation between triangulations of $conv(A)$ and cubical tilings of $Z(A)$,</li> <li>indicate why, and in which context, one of both concepts should be more natural than the other.</li> </ul> <p>p.s. may be someone reputed enough would like to create the tag oriented-matroids?</p> http://mathoverflow.net/questions/93772/triangulations-of-polytopes-and-tilings-of-zonotopes/122515#122515 Answer by Camilo Sarmiento for Triangulations of polytopes and tilings of zonotopes Camilo Sarmiento 2013-02-21T07:58:36Z 2013-02-21T08:06:15Z <p>Triangulations of polytopes are "more fundamental" than cubical tilings of zonotopes. By the <a href="http://www.ams.org/mathscinet-getitem?mr=1763304" rel="nofollow">Cayley trick</a>, 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. </p> <p>This can be seen as an addition to the <a href="http://www.ams.org/mathscinet-getitem?mr=1310586" rel="nofollow">Bohne-Dress theorem</a>, relating tilings of a zonotope to liftings of the associated oriented matroid.</p>