A note on the convexity assumption of possibilistic correlation

In 2005, Carlsson, Fullér and Majlender introduced a measure of possibilistic correlation   of fuzzy numbers AA and BB by their joint possibility distribution  CC as an average degree of interaction   between the γγ-level sets of AA and BB as compared to their individual dispersions. They proved that this possibilistic correlation coefficient   can never exceed 11 in absolute value, if all γγ-level sets of the joint possibility distribution CC are convex.In this communication, we shall formulate a special class of joint possibility distributions with non-convex γγ-level sets, for which the correlation coefficient can take values outside the interval [−1,1][−1,1]. In particular, this result will show that the assumption about the convexity of the level sets of CC is essential for the possibilistic correlation to be bounded by the interval [−1,1][−1,1].