Properties and in Orlicz spaces and some of their consequences

We introduce a new geometric property and we show that it is equivalent to its uniform version . Next we introduce a new property and we show that if a separable Banach space X has this property, then both X and X⁎ have the weak fixed point property. We also prove that a uniformly Gateaux differentiable Banach space has property and that if X⁎ has property , then X has the (UKK)-property. Criteria in order that Orlicz spaces have the properties , and (NUS⁎) are given.