  We show that any smooth lattice polytope P with codegree greater or equal
than (dim(P)+3)/2 (or equivalently, with degree smaller than dim(P)/2), defines
a dual defective projective toric manifold. This implies that P is Q-normal (in
the terminology of a recent paper by Di Rocco, Piene and the first author) and
answers partially an adjunction-theoretic conjecture by Beltrametti and
Sommese. Also, it follows that smooth lattice polytopes with this property are
precisely strict Cayley polytopes, which completes the answer of a question of
Batyrev and the second author in the nonsingular case.
