Rotund Points, Nested Sequence of Balls and Smoothness in Banach Spaces

In this work, we continue our study of rotund points and, inter alia, show that various characterizations of strict convexity of X* like Vlasov's Theorem or Taylor-Foguel Theorem are locally consequences of properties of rotund points- a notion strictly stronger than extreme points. We observe that rotund points are a special kind of exposed points and obtain similar characterizations of wALUR and ALUR points. We characterize smooth, very smooth and Frechet smooth points in terms of straight unbounded nested sequences of balls. Various related notions are also studied in the local form.