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 f : U \ A → R can be uniquely extended (with preservation of preconvexity) onto U . In fact we show that a more general version of these result holds for semiconvex functions.