Near convexity, near smoothness and approximative compactness of half spaces in Banach spaces

Some Metric and Topological Properties of Nearly Strongly and Nearly Very Convex Spaces

“…Then we have LUR ⇒ strongly convex ⇒ nearly strongly convex ⇒ nearly very convex, and these four concepts are different (see [ZL11, Examples 2.5, 2.6, and 2.7]) (for examples outside the context of reflexive spaces -all of them nearly very smoothconsider [Dr14, Theorem 1], where it is proved that every infinite-dimensional Banach space with separable dual admits an equivalent wLUR norm which is not LUR: It is obvious that every wLUR space is nearly very convex; this wLUR equivalent norm cannot be nearly strongly convex, since this last property implies property (H) that, together with wLUR, implies LUR, see below). The concepts nearly strongly convex and nearly very convex are discussed, e.g., in [BLLN08], [FW01], [GM11], [ZL11], [ZL12], [ZMLG15], and [ZS09], and they are related to questions of approximation in Banach spaces. We may mention, for example, a characterization of nearly strict convexity in terms of the preduality mapping:…”