Abstract
The $$d$$ -dimensional simplicial, terminal, and reflexive polytopes with at least $$3d-2$$ 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 Øbro (Manuscr Math 125(1): 69–79, 2008). Smooth Fano polytopes play a role in algebraic geometry and mathematical physics.
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