Consider an $n$-simplex with vertices given by $(0,0,\dots,r_i,r_{i+1},\dots,r_n)$ where $r_1,\dots,r_n$ are given natural numbers, and $i=0,1,\dots,n+1$.

Does this simplex admit a regular, unimodular triangulation?

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    $\begingroup$ Maybe, $r_0$ is also positive integer? $\endgroup$ – Fedor Petrov Apr 8 '16 at 5:25
  • $\begingroup$ @FedorPetrov Ah, yes. $\endgroup$ – Per Alexandersson Apr 8 '16 at 12:31
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    $\begingroup$ I am a little late to the party, but what is a regular unimodular triangulation? $\endgroup$ – Igor Rivin Apr 3 '17 at 13:41
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    $\begingroup$ @IgorRivin I am a little late to answer, but unimodular means onto simplices with integer vertices and volume $1/n!$; regular means that there exists a convex function on the whole simplex which is affine on each simplex of the triangulation and has different gradients on any two of them. See Gelfand Kapranov Zelevinsky. $\endgroup$ – Fedor Petrov Dec 25 '18 at 4:47

The answer is yes. Consider the hyperplanes $x_i=x_j+m$ and $x_i=m$ for all $i$ $j$ and integers $m$. These hyperplanes triangulate the entire space into unimodular simplices, and the hyperplanes are compatible with the hyperplanes that determine the polytope in the question (meaning each face is determined by an intersection of some of these hyperplanes). This induces a triangulation of the polytope, and it should be clear that this is regular.

EDIT: This is not a correct answer to the question as stated, I had a different question in mind. I was wondering if Gelfand-Tsetlin polytopes admits unimodular triangulations (which they do), and the original question I had in mind (but did not write correctly) would have implied that.

  • $\begingroup$ This doesn't seem right to me. What if the vertices are (0,0), (0,2), and (3,2)? The edge joining (0,0) to (3,2) is not an intersection of lines such as you describe. $\endgroup$ – Hugh Thomas Nov 14 '14 at 17:02
  • $\begingroup$ Ah, you are right! I realized I asked the wrong question. This is the solution to the problem I had in mind, I should edit one of them... Or perhaps delete both... $\endgroup$ – Per Alexandersson Nov 14 '14 at 17:05
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    $\begingroup$ I think the question as stated is a perfectly good question, so I would be in favour of not deleting it. I would also be interested in knowing what question you meant to ask, though I guess it should be a separate question. Also (though this may become clear when I know what question you were answering) it isn't clear to me why the triangulation you give in your answer is automatically regular. $\endgroup$ – Hugh Thomas Nov 14 '14 at 17:16
  • $\begingroup$ Yeah, it is not clear to me either, actually, now when you mention it... but somehow, the triangulation is so nice, so that I would find it difficult not to be in some sense.. but yes, this is a gap. $\endgroup$ – Per Alexandersson Nov 14 '14 at 23:09
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    $\begingroup$ Any triangulation (or subdivision) defined by hyperplane cuts, such as the one in the answer, is regular. The sum of distances to the hyperplanes is a valid height function. $\endgroup$ – Francisco Santos Feb 17 '15 at 9:24

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