Extensions of convex and semiconvex functions and intervally thin sets

We call A ⊂ R N intervally thin if for all x , y ∈ R N 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 R N 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 this result holds for semiconvex functions.