On the existence of aggregation functions with given super-additive and sub-additive transformations

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 $A^*=g$.

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