The rise and fall of L-spaces

Abstract

For a quite long period, the so-called L-structure or L-spaces have been studied by some authors. They have several trivial misconceptions such as their L-spaces extend the well-known generalized convex (G-convex) spaces. In order to clarify this matter and others, we show that our KKM theory on abstract convex spaces improves typical results in L-spaces. Main topics in this paper are related to extensions of the Himmelberg fixed point theorem. Since such studies are beyond of L-spaces, we cordially claim that now is the proper time to give up the useless study on L-spaces and their variants FC-spaces.