On approximately additive functions

Let C be a convex symmetric subset of a real Banach space F and K be a subgroup of the group ( F , + ) . Let E be a real linear space, h : E → F , and h ( x + y ) − h ( x ) − h ( y ) ∈ K + C for x , y ∈ E . We prove that under some additional assumptions h can be represented in the form: h = A + γ + κ with an additive (or linear) A : E → F and some γ : E → C , κ : E → K .