Minimal non-convex words

Using a combinatorial characterization of digital convexity based on words, one defines the language of convex words. The complement of this language forms an ideal whose minimal elements, with respect to the factorial ordering, appear to have a particular combinatorial structure very close to the Christoffel words. In this paper, those words are completely characterized as those of the form uwkv where k≥1, w=u⋅v and u,v,w are Christoffel words. Also, by considering the most balanced among the unbalanced words, we obtain a second characterization for a special class of minimal non-convex words that are of the form u2v2 corresponding to the case k=1 in the previous form.