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If I understand your question correctly, you're saying that the given information is the face structure of a 3-dimensional convex polytope, and you would like a subdivision of the polytope into tetrah …

Edited to add: I now think the answer below is completely wrong. The three-dimensional cyclohedron has 12 facets, while the three-dimensional polytope the OP is looking for should have 14. This is t …

Are there any purely combinatorial criteria which allow you to deduce that a spherical simplicial complex is polytopal (i.e., there exists a simplicial polytope whose boundary is isomorphic to it)? F …

Yes. Let $\Sigma$ be the fan corresponding to $P$. Section 2.6 of Fulton's "Introduction to Toric Varieties" explains how to perform toric resolution of singularities on $\Sigma$ so as to produce a fa …

The fan Hales is using is called the "face fan" of the polytope. In toric varieties, one mainly considers the outer normal fan of a polytope, which has a ray for each facet (perpendicular to it). …

What happens as you push $v$, is that you can break a face into two. Start with a cube in $\mathbb R^3$, choose a vertex, and start to push it into the interior of the cube. Each of the three facets …

There is a canonical thing to do if you have a lattice quotient of weak order on the symmetric group. Namely, you take the Coxeter fan (which has one maximal cone for each element of $S_n$) and you g …

Proposition 4 of Morelli's paper "Pick's Theorem and the Todd class of a toric variety" gives a sufficient condition: it describes a setting in which there is a positive formula for coefficient of $x^ …

Let V be the set of vertices, P their convex hull, and Q the polytope defined by the hyperplanes. If P is assumed to be simple, it suffices to check that each x in V is also a vertex of Q. By your …

Let's consider the associahedron whose vertices are triangulations of an $n$-gon. Sleator, Tarjan, and Thurston (Rotation distance, triangulations, and hyperbolic geometry, JAMS, 1988) give a simple a …

Let's think of choosing the values of $f$ at the grid points one grid point at a time, in such a way that the result is convex. If you work systematically through the grid points in an order so t …

The righthand side is clearly contained in the lefthand side. We now want to show the lefthand side (i.e., $Q_I$) is contained in the righthand side. Since $Q_I$ is the convex hull of its lattice poin …