Ehrhart polynomials of lattice polytopes with normalized volumes 5 (Q1713788)

Classifying lattice polytopes, that is, polytopes with all vertices in \(\mathbb{Z}^d\), up to unimodular equivalence is one of the major problems in discrete geometry. Due to the problem's complexity one inevitably has to restrict to classes described by small natural parameters. Common approaches include: small dimension and small volume; small number of contained lattice points; small normalized volumes but arbitrary dimension. The author follows the latter approach and contributes a classification of the \(\delta\)-vectors of lattice polytopes of normalized volume at most \(5\). Here, the \(\delta\)-vector \(\delta(P)=(\delta_0,\delta_1,\ldots,\delta_d)\) of a lattice polytope \(P \subseteq \mathbb{R}^d\) is defined as the coefficient vector of the polynomial \[\delta(P,t) = (1-t)^{d+1} \left(1 + \sum_{n\geq 1}\#(nP \cap \mathbb{Z}^d) \cdot t^n\right).\] The \(\delta_i\) are all non-negative integers and the normalized volume of \(P\) is given by \(\delta_0+\delta_1+\ldots+\delta_d\). The main result is based on the already established classification of \(\delta\)-vectors of lattice simplices with normalized volume \(5\) [\textit{A. Higashitani}, Integers 14, Paper A45, 15 p. (2014; Zbl 1384.11041)], paired with the following observation that should be interesting and useful in its own right: Every lattice polytope whose normalized volume is a prime, is either an empty simplex or spanning (Theorem 1.1 in the paper).