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

A Simple Combinatorial Criterion for Projective Toric Manifolds with Dual Defect