Unimodular Smooth Fano Polytopes and their Relation with Ewald Conditions

Smooth Fano polytopes (SFP) play an important role in toric geometry and combinatorics. In this paper, we introduce a specific subcollection of them, i.e., the unimodular smooth Fano polytopes (USFP). In Section 2, they are verified to satisfy the three (weak, strong, star) Ewald conditions. Besides, a characterisation of USFPs is provided as a corollary of the famous Seymour's decomposition theorem. Then, we briefly introduce the works by Luis Crespo on deeply monotone polytopes and give a proof of the claim that any deeply monotone polytope is in fact the dual polytope of some USFP. In other words, we extend his results on deeply monotone polytopes to the case of USFPs.