  In the hierarchy of structural sophistication for lattice polytopes, normal
polytopes mark a point of origin; very ample and Koszul polytopes occupy bottom
and top spots in this hierarchy, respectively. In this paper we explore a
simple construction for lattice polytopes with a twofold aim. On the one hand,
we derive an explicit series of very ample 3-dimensional polytopes with
arbitrarily large deviation from the normality property, measured via the
highest discrepancy degree between the corresponding Hilbert functions and
Hilbert polynomials. On the other hand, we describe a large class of Koszul
polytopes of arbitrary dimensions, containing many smooth polytopes and
extending the previously known class of Nakajima polytopes.
