Ehrhart polynomial roots of reflexive polytopes

Recent work has focused on the roots z∈C of the Ehrhart polynomial of a lattice polytope P. The case when Rz=−1/2 is of particular interest: these polytopes satisfy Golyshev's "canonical line hypothesis". We characterise such polytopes when dim(P)≤7. We also consider the "half-strip condition", where all roots z satisfy −dim(P)/2≤Rz≤dim(P)/2−1, and show that this holds for any reflexive polytope with dim(P)≤5. We give an example of a 10-dimensional reflexive polytope which violates the half-strip condition, thus improving on an example by Ohsugi–Shibata in dimension 34.