A Schur-concave shuffle of M outside the closed convex hull of associative copulas

Document Type : Original Manuscript

Author

Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology, Bratislava, Slovakia

10.22111/ijfs.2026.55340.9816

Abstract

The class of associative copulas is uniformly compact but not convex, and every associative copula is Schur-concave and  commutative. Two open problems posed by Durante at FSTA 2010 concern its closed convex hull and the extreme points  of that hull. We give a negative answer to the first problem by constructing an explicit family of Schur-concave  commutative copulas outside the hull. Each member is a symmetric shuffle of the upper Fr'{e}chet-Hoeffding bound. We  compute its exact uniform distance from the hull using a star-shapedness property of antidiagonal sections of associative  copulas. For the second problem, we show that every extreme point of the hull is associative and reformulate the  remaining inclusion question using associative copulas that are extreme among all copulas. We prove that ordinal sums  of the lower Fr'{e}chet-Hoeffding bound form a proper subclass of these copulas and generate a strictly smaller closed  convex hull. Nevertheless, the independence copula admits an integral representation over such ordinal sums and is not  extreme in the hull of associative copulas. 

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References

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Iranian Journal of Fuzzy Systems
Volume 23, Issue 6
November and December 2026
Pages 223-238

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