A Simple Combinatorial Criterion for Projective Toric Manifolds with Dual Defect

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.