  In this note we study restrictions on the recently introduced super-additive
and sub-additive transformations, $A\mapsto A^*$ and $A\mapsto A_*$, 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
$f\le g$ sub-additive and super-additive on $[0,\infty[^n$, respectively, with
zero value at the origin and satisfying relatively mild extra conditions, for
which there exists no aggregation function $A$ on $[0,\infty[^n$ such that
$A_*=f$ and $A^*=g$.
