The role of generalized convexity in conic copula constructions

Inspired by the notion of a conic semi-copula, we introduce upper conic, lower conic and biconic semi-copulas with a given section. Such semi-copulas are constructed by linear interpolation on segments connecting the graph of a strict negation operator to the points (0,0) and/or (1,1). The important subclasses of upper conic, lower conic and biconic (quasi-)copulas with a given section are characterized. The notion of generalized convexity turns out to play a key role in this characterization. Some examples are also provided.