Approximation of convex bodies by polytopes corresponding to smooth toric varieties

Question

Let $P\subset \mathbb{R}^n$ be an $n$-dimensional polytope with rational vertices. There is a well known construction which produces an $n$-dimensional algebraic variety $X_P$ called toric variety. In general $X_P$ may not be smooth, but the smoothness condition imposes some well known restrictions on $P$. (For example, a necessary condition is that $P$ has to be simple, i.e. any vertex has exactly $n$ adjacent edges.)

Is it true that any convex compact set in $\mathbb{R}^n$ can be approximated in the Hausdorff metric by rational polytopes for which the corresponding toric varieties are smooth?

Answer 1

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 fan $\Sigma'$ whose toric variety is smooth and which refines $\Sigma$. This can be done in such a way that each step corresponds to adding a ray in the interior of some cone $C$ of $\Sigma$ and subdividing the faces of $\Sigma$ that include $C$ as necessary so that what you have stays a fan. (For instance, you can subdivide all the maximal-dimensional cones, then all those of dimension one less, and so on, until you have produced the barycentric subdivision of the original fan, which is simplicial. Then follow the instructions in Fulton as to how to subdivide further so as to get a smooth fan.)

On the polytope side, the step of adding a ray in the interior of a face is equivalent to introducing a new hyperplane that shaves off the face corresponding to $C$. This can be done in a way that only modifies $P$ within a distance $\epsilon$ of $C$. Since this applies to each step, it also applies to the whole process.