For a long period, the so-called L-structure or L-spaces have been studied by several authors. However, there exist several misconceptions, such as the belief that L-spaces extend the well-known generalized convex (G-convex) spaces. In order to clarify these misunderstandings, we demonstrate that KKM theory on abstract convex spaces improves typical results in L-spaces. The main topics in this paper relate to extensions of the Himmelberg fixed point theorem. Since such studies transcend the framework of L-spaces, we assert that it is time to abandon the unproductive study of L-spaces and their variants, FC-spaces.