In this note we study restrictions on the recently introduced super-additive and sub-additive transformations, A → A∗ and A → A∗, of an aggregation function A. We prove that if A∗ has a slightly stronger property of being strictly directionally convex, then A = A∗ and A∗ is linear; dually, if A∗ is strictly directionally concave, then A = A∗ and A∗ is linear. This implies, for example, the existence of pairs of functions f≤g sub-additive and super-additive on [0, ∞[n, respectively, with zero value at the origin and satisfying relatively mild extra conditions, for which there exists no aggregation function A on [0, ∞[n such that A∗=f and A∗=g
