Convex lineability in copula and quasi-copula sets

In this paper, we investigate several subsets of $n$-copulas and $n$-quasi-copulas from the perspective of convex-lineability and the recently introduced concept of convex-spaceability. Our purpose is to determine when such families contain extremely large algebraic structures, namely linearly independent sets of cardinality of the continuum whose convex hull, and in some cases a closed convex linearly independent subset, remain entirely inside the class under study. These include the families of asymmetric copulas, copulas with maximal asymmetric measure, and proper $n$-quasi-copulas, among others. In contrast, for several other natural classes of copulas we show that (maximal) convex lineability holds while convex spaceability remains an open problem.

💡 Research Summary

The paper investigates the presence of large algebraic and geometric structures within subsets of multivariate copulas and quasi‑copulas, focusing on the notions of convex‑lineability and the newly introduced convex‑spaceability. Classical lineability, which seeks infinite‑dimensional vector subspaces inside a given set, is unsuitable for copulas because multiplication by negative scalars or arbitrary constants destroys the boundary conditions that define a copula. To overcome this, the authors adopt convex‑lineability: a set M is convex‑lineable if there exists an infinite linearly independent family A such that the convex hull conv(A) is contained in M. When the cardinality of A equals that of the ambient space (the continuum), the set is called maximally convex‑lineable. Convex‑spaceability strengthens this by requiring a closed convex set that contains an infinite linearly independent subset, again possibly maximal.

Working in the Banach space of n‑copulas on