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?**