In recent years, Answer Set Programming (ASP), logic programming under the stable model or answer set semantics, has seen several extensions by generalizing the notion of an atom in these programs: be it aggregate atoms, HEX atoms, generalized quantifiers, or abstract constraints, the idea is to have more complicated satisfaction patterns in the lattice of Herbrand interpretations than traditional, simple atoms. In this paper we refer to any of these constructs as generalized atoms. Several semantics with differing characteristics have been proposed for these extensions, rendering the big picture somewhat blurry. In this paper, we analyze the class of programs that have convex generalized atoms (originally proposed by Liu and Truszczynski in ) in rule bodies and show that for this class many of the proposed semantics coincide. This is an interesting result, since recently it has been shown that this class is the precise complexity boundary for the FLP semantics. We investigate whether similar results also hold for other semantics, and discuss the implications of our findings.