Extensions of convex and semiconvex functions and intervally thin sets

We call A⊂RN intervally thin if for all x,y∈RN and ε>0 there exist x′∈B(x,ε), y′∈B(y,ε) such that [x′,y′]∩A=∅. Closed intervally thin sets behave like sets with measure zero (for example such a set cannot “disconnect” an open connected set). Let us also mention that if the (N−1)-dimensional Hausdorff measure of A is zero, then A is intervally thin. A function f is preconvex if it is convex on every convex subset of its domain. The consequence of our main theorem is the following: Let U be an open subset of RN and let A be a closed intervally thin subset of U. Then every preconvex function can be uniquely extended (with preservation of preconvexity) onto U. In fact we show that a more general version of this result holds for semiconvex functions.