By the continuity of preduality map, we give some necessary and sufficient conditions of the strongly convex and very convex spaces, respectively. Using nearly strong convexity of X, we give some equivalent conditions that every element in X is strongly unique of order p, bounded strongly unique of order p, and locally strongly unique of order p.

Five counterexamples are given, which show relations among the new convexities and some important convexities in Banach space. Under the assumption that Banach space is nearly very convex, we give a sufficient condition that bounded, weakly closed subset of has the farthest points. We also give a sufficient condition that the farthest point map is single valued in a residual subset of when is very convex.

Montesinos Santalucia, V.; Liu, CY.; Gong, WZ. (2015). Geometric properties and continuity of the pre-duality mapping in Banach space. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas. 109 (2) AbstractWe use the preduality mapping in proving characterizations of some geometric properties of Banach spaces. In particular, those include nearly strongly convexity, nearly uniform convexity -a property introduced by K. Goebel and T. Sekowski-, and nearly very convexity.