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Finding the exact number of unimodular triangulations of a cube in higher dimension is, I would say, out of reach. In fact, I would say the number is doubly exponential in n (that is, exponential in t …

Here is a lower bound of $2^{\Omega(2^n)}$ for the number of unimodular triangulations. Let me start with the triangulation described by Włodzimierz Holsztyński, which is indeed quite classical and I …

I have just seen this question, while looking for something else. The answer to 1 is indeed "no", and an explicit proof appears in Lemma 2.2 of Ceballos, Santos, and Ziegler - Many non-equivalent real …

As pointed out by Sam Hopkins in a comment above, secondary polytopes can be seen as a particular case of the fiber polytopes of Billera and Sturmfels (https://doi.org/10.2307/2946575). This fiber pol …

Gale transforms allow to settle the case of odd $d$ and even $v$ completely, in the positive (Matteo's answer in this case needed $v\ge 2d$). Remember that the Gale transform of a $d$-polytope with $v …

The answer is YES in the following stronger form: Lemma 1: Let $d>n$ and let $p\in dP\cap \mathbb{Z}^n$. Then there is an $i\in\{1,\dots, n\}$ and points $p'\in iP\cap \mathbb{Z}^n$, $p_1,\dots, p_{d …

(Edited Feb 3 to correct previous wrong example) Let me try a simpler example than in my comments. Consider the lattice $\mathbb{Z}^2$ and the polygon $P$ with vertices $(0,0)$, $(1,-3)$, $(2,-3)$, $ …

The conditions you pose on $P_0$ imply that it is a reflexive polytope. (That is, a lattice polytope with the origin in its interior and such that its polar dual is also a lattice polytope). There are …

Every simplicial polytope is rational, hence there are infinitely many rational polytopes in any fixed dimension. In contrast, there are finitely many 0/1 polytopes, since they cannot have more than $ …

The answer to 2 and 1 is "No". For (2), observe that a positive answer here would imply all Delaunay simplices to be unimodular (i.e., have volume equal to $1/n!$ times the volume of a fundamental p …

Unless you have some additional structure on the points $A_1$,...,$A_n$ your problem is as hard (and as easy) as computing the convex hull of $2n$ points in $\mathbb{R}^n$, and triangulating this conv …

I think the answer is yes and the following is a sketch of proof: The second barycentric subdivision of a $\Delta$-complex is a triangulation. The f-vector of a $\Delta$-complex and of its barycentr …

The answer is "never" (except in the obvious case $d=2$, $n\ge 4$). $C(n,d)$ is neighborly, meaning that every $d/2$ or less vertices define a simplex. In particular, for $d\ge 4$ its graph is comple …

I think the whole list of Gjiergji (and lightly more, see below) follows from Wythoffian operations. For the cube-octahedron family, for example, consider the following diagram: The polyhedron along a …