Properties and in Orlicz spaces and some of their consequences

We introduce a new geometric property ( A ˜ 2 ) and we show that it is equivalent to its uniform version ( U A ˜ 2 ) . Next we introduce a new property ( U A ˜ 2 ) ⁎ 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 ( U A ˜ 2 ) and that if X ⁎ has property ( U A ˜ 2 ) ⁎ , then X has the ( UKK ) -property. Criteria in order that Orlicz spaces have the properties ( U A ˜ 2 ) , ( U A ˜ 2 ) ⁎ and ( NUS ⁎ ) are given.