Several Geometric Properties in Banach Spaces and Their Further Application in Orlicz Spaces

In this paper, locally nearly uniformly convex (LNUC) is studied in Banach space. Furthermore, the implication relationship between (LNUC) and the Kadec–Klee property (KK), the fixed–point property (FPP) are investigated in Banach space. Finally, the relationship between the uniform Kadec−Klee property (UKK), the coordinate-wise uniform Kadec–Klee property (UKKC), the coordinate-wise Kadec–Klee property (Hc) and δ2 conditions are investigated in Orlicz sequence spaces equipped with the Orlicz norm, meanwhile we get a criteria that Orlicz sequence spaces equipped with the Orlicz norm are (LNUC).