On removable sets for convex functions

In the present article we provide a sufficient condition for a closed set F∈RdF∈Rd to have the following property which we call c  -removability: Whenever a continuous function f:Rd→Rf:Rd→R is locally convex on the complement of F  , it is convex on the whole RdRd. We also prove that no generalized rectangle of positive Lebesgue measure in R2R2 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⊂RdF⊂Rd is such that any locally convex function defined on Rd∖FRd∖F has a unique convex extension on RdRd. 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 R2R2.