  Q-factorial Gorenstein toric Fano varieties X of dimension d with Picard
number rho(X) correspond to simplicial reflexive d-polytopes with rho(X)+d
vertices. Casagrande showed that any simplicial reflexive d-polytope has at
most 3d vertices, if d is even, respectively, 3d-1, if d is odd. Moreover, it
is known that equality for d even implies uniqueness up to unimodular
equivalence. In this paper we completely classify all simplicial reflexive
d-polytopes having 3d-1 vertices, corresponding to d-dimensional Q-factorial
Gorenstein toric Fano varieties with Picard number 2d-1. For d even, there
exist three such varieties, with two being singular, while for d odd (d > 1)
there exist precisely two, both being nonsingular toric fiber bundles over the
projective line. This generalizes recent work of the second author.
