A note on the super-additive and sub-additive transformations of aggregation functions: the multi-dimensional case. (English) Zbl 1424.26032

Summary: For an aggregation function \(A\) we know that it is bounded by \(A^*\) and \(A_*\) which are its super-additive and sub-additive transformations, respectively. Also, it is known that if \(A^*\) is directionally convex, then \(A=A^*\) and \(A_*\) is linear; similarly, if \(A_*\) is directionally concave, then \(A=A_*\) and \(A^*\) is linear. We generalize these results replacing the directional convexity and concavity conditions by the weaker assumptions of overrunning a super-additive function and underrunning a sub-additive function, respectively.


26B40 Representation and superposition of functions
26B30 Absolutely continuous real functions of several variables, functions of bounded variation
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