  We show that h*-vectors of alcoved polytopes P in R^n (of Lie type A) are
unimodal if they contain interior lattice points and their facets have lattice
distance 1 to the set of interior lattice points. The maximal possible such
distance for general alcoved polytopes is shown to be dim(P)-1. A secondary
purpose of the paper is to serve as a guide to previous work surrounding
unimodality of h*-vectors of alcoved polytopes and related questions.
