# Regular unimodular triangulation for a certain simplex

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?

• Maybe, $r_0$ is also positive integer? – Fedor Petrov Apr 8 '16 at 5:25
• @FedorPetrov Ah, yes. – Per Alexandersson Apr 8 '16 at 12:31
• I am a little late to the party, but what is a regular unimodular triangulation? – Igor Rivin Apr 3 '17 at 13:41
• @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. – 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.