On Barycentric transformations of Fano polytopes

We introduce the notion of barycentric transformation of Fano polytopes, from which we can assign a certain type to each Fano polytope. The type can be viewed as a measure of the extent to which the given Fano polytope is close to be K"ahler-Einstein. In particular, we expect that every K"ahler-Einstein or symmetric Fano polytope is of type $B_{\mathrm{\infty }}$. We verify this expectation for some low dimensional cases. We emphasize that for a Fano polytope $X$ of dimension $1$, $3$ or $5$, $X$ is K"ahler-Einstein if and only if it is of type $B_{\mathrm{\infty }}$.