Geodesic
Nonconvex Lipschitz function in plane which is locally convex outside a discontinuum
Commentationes Mathematicae Universitatis Carolinae, Tome 55 (2014) no. 4, pp. 509-521.

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We construct a Lipschitz function on $\mathbb R^2$ which is locally convex on the complement of some totally disconnected compact set but not convex. Existence of such function disproves a theorem that appeared in a paper by L. Pasqualini and was also cited by other authors.
Classification : 26B25, 52A20, 52A41
Keywords: convex function; convex set; exceptional set 
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     title = {Nonconvex {Lipschitz} function in plane which is locally convex outside a discontinuum},
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Pokorný, Dušan. Nonconvex Lipschitz function in plane which is locally convex outside a discontinuum. Commentationes Mathematicae Universitatis Carolinae, Tome 55 (2014) no. 4, pp. 509-521. http://geodesic.mathdoc.fr/item/CMUC_2014__55_4_a7/