Number of vertices of a lattice reflexive polytope

Question

Casagrande's paper "The number of vertices of a Fano polytope" says that for $P$ a simplicial reflexive polytope of dimension $n$ has no more than $3n$ vertices.

The polymake database classifies 124 smooth reflexive 4-dimensional lattice polytopes (smooth implies simplicial), and there are several smooth reflexive lattice polytopes with, say $\geq 24$ vertices. How are there smooth reflexive lattice polytopes in the polymake database with $\geq 24$ vertices if the upper bound is supposed to be $3n$?

Answer 1

This is due to a confusion about the notion of "smooth polytopes" in the literature. Casagrande's bound applies to simplicial polytopes. Reflexive polytopes come in pairs where -- if this pair classifies a smooth projective toric Fano variety -- exactly one is the polytopes is simplicial, and the other simple. The polytopes in the polymake "smooth reflexive lattice polytope" database are actually simple, rather than simplicial: For instance, Casagrande's bound applies instead to the number of facets of the polytopes in the polymake database.