Unimodular defect and antipodal lattice points of smooth reflexive polytopes

Source abstract

Let $P$ be a smooth reflexive polytope and let $Q=P^{*}$ be the dual of smooth Fano polytope. Ewald's conjecture (1988) predicts that the lattice points of $P\cap (-P)$ contain a lattice basis. In general it is not even known whether $P\cap (-P)$ contains a nonzero lattice point; we call this weakest form of the conjecture the antipodal condition. We define the unimodular defect of a finite set of lattice vectors as the least number of vectors that must be removed to leave a unimodular set, and the unimodular defect of $Q$ as that of its vertex set. Defect zero gives the unimodular smooth Fano polytopes, for which Ewald's conjecture is known. Our main result is that every smooth Fano polytope of unimodular defect at most two has a \emph{flat ridge}, a ridge $W$ with $|\det(v,W)|\leq 1$ for all vertices $v$; consequently its dual satisfies the antipodal condition. We also consider removed sets of pairwise non-adjacent vertices. Using results of Nill on independent vertices, we show that every such set has at most two elements unless $Q$ is unimodular, so the antipodal condition holds whenever such a set exists.