Smooth Fano Polytopes with Many Vertices

The $$d$$ d -dimensional simplicial, terminal, and reflexive polytopes with at least $$3d-2$$ 3 d - 2 vertices are classified. In particular, it turns out that all of them are smooth Fano polytopes. This improves on previous results of Casagrande (Ann Inst Fourier (Grenoble) 56(1):121---130, 2006) and Obro (Manuscr Math 125(1): 69---79, 2008). Smooth Fano polytopes play a role in algebraic geometry and mathematical physics.