Regular lattice polytopes and root systems

Consider Lambda, a lattice in a real finite-dimensional vector space. Here we are interested in lattice polytopes, that is, convex polytopes with vertices in Lambda. Consider the group G of the affine real transformations that map the lattice onto itself. Replacing the group of Euclidean motions by the group G one can define the notion of regular lattice polytopes. More precisely, a lattice polytope is said to be regular if the subgroup of G which preserves the polytope acts transitively on the set of its complete flags. In this paper, we associate to each regular lattice polytope a root system. This association allows us to give a new proof of the classification of regular lattice polytopes recently obtained by Karpenkov.