In general topology, connectedness is a fundamental concept. Convex structures, analogous to topological structures, enable the generalization of numerous properties from topological spaces. In this paper, we first define $r$-connectivity for $L$-subsets in convex $(L,M)$-fuzzy hull spaces based on convex $(L,M)$-fuzzy hull operators. We then establish several equivalent characterizations of $r$-connected $L$-subsets and explore their key properties. In particular, by leveraging these operators, we introduce the finite product property of $r$-connecivity for product spaces within the framework of convex $(L,M)$-fuzzy hull spaces.

This study is carried out in the context of $(L,M)$-fuzzy weak hull operators. First, we introduce the concept of an $(L,M)$-fuzzy quasi-hull operator and prove that $(L,M)$-fuzzy quasi-hull operators are categorically isomorphic to $(L,M)$-fuzzy supratopologies. Second, by imposing a compatibility condition with group structures, we propose the notion of an $(L,M)$-fuzzy weak hull group and further investigate its relevant properties, such as those related to its subgroups and product groups.