Characterizing Schur-concave commutative copulas as the closure of associative ones

Let ${\mathcal{C}}_a$ denote the class of associative copulas, and let ${\bar{\mathcal{C}}}_a$ be the closure, in the uniform metric $d_{\mathrm{\infty }}$, of the convex hull of ${\mathcal{C}}_a$ . It is known that ${\mathcal{C}}_a\subseteq {\mathcal{C}}_{SC}$, the class of Schur-concave commutative copulas. We prove the reverse inclusion, establishing ${\bar{\mathcal{C}}}_a={\mathcal{C}}_{SC}$.