Unimodular Smooth Fano Polytopes and their Relation with Ewald Conditions

TL;DR

This paper introduces unimodular smooth Fano polytopes (USFP), verifies their Ewald conditions, characterizes them via Seymour's decomposition, and links them to deeply monotone polytopes, extending existing results.

Contribution

It defines USFPs, proves they satisfy Ewald conditions, characterizes them using Seymour's decomposition, and connects them to deeply monotone polytopes, extending prior work.

Findings

USFP satisfy all three Ewald conditions

USFP are characterized by Seymour's decomposition theorem

Deeply monotone polytopes are dual to USFPs

Abstract

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.