The key role of convexity in some copula constructions

We start with some binary (“outer”) copula, apply it to an arbitrary binary (“inner”) copula and its dual (the latter being transformed by some real function) and ask under which conditions the result is again a binary copula. Sufficient convexity conditions for the transformation function and for the “outer” copula (ultramodularity and Schur concavity) are given, thus generalizing the scenarios considered in Klement et al. (J Math Inequal 11(2):361381, 2017) and Manstavičius and Bagdonas (Fuzzy Sets Syst 354:4862, 2019). In general, these sufficient conditions are not necessary (and a counterexample is provided), but in some distinguished cases necessary and sufficient conditions can be given. Several well-known families of copulas can be obtained in this way. We also present a few extensions for special “outer”/“inner” copulas and/or transformation functions, as well as some counterexamples