Locally Nearly Uniformly Convex Points in Orlicz Spaces Equipped with the Luxemburg Norm

This research explores two novel geometric concepts—nearly convex points and locally nearly uniformly convex points within the frameworks of Banach spaces and Orlicz spaces equipped with the Luxemburg norm. First, we establish the general characterization criteria for nearly convex points in Banach spaces. Then, we analyze the intrinsic connection between locally nearly uniformly convex points and nearly extreme points in Banach spaces. Additionally, we provide comprehensive characterizations of locally nearly uniformly convex points in both Orlicz function spaces and Orlicz sequence spaces under the Luxemburg norm. These findings enrich the geometric theory system of Banach and Orlicz spaces, offering new theoretical support for related research directions.