On removable sets for convex functions

In the present article we provide a sufficient condition for a closed set F ∈ R d to have the following property which we call c-removability: Whenever a continuous function f : R d → R is locally convex on the complement of F, it is convex on the whole R d . We also prove that no generalized rectangle of positive Lebesgue measure in R 2 is c-removable. Our results also answer the following question asked in an article by Jacek Tabor and Józef Tabor (2010) [5]: Assume the closed set F ⊂ R d is such that any locally convex function defined on R d ∖ F has a unique convex extension on R d . Is F necessarily intervally thin (a notion of smallness of sets defined by their “essential transparency” in every direction)? We prove the answer is negative by finding a counterexample in R 2 .