On PropertyŽM.and Its Generalizations

Properties strictŽM.and uniformŽM.are introduced. It is shown that if X has property ŽM. and is uniformly convex in every direction, then X has both strict ŽM. and uniform ŽM.. It is also shown that if X* is separable, then strict ŽM. implies uniform ŽM. and property ŽM. implies weak uniform normal structure. Relations with other geometrical properties of Banach spaces are also discussed.