Characterizations and applications of three types of nearly convex points

Abstract

By using some geometric properties and nested sequence of balls, we prove seven necessary and sufficient conditions such that a point $x$ in the unit sphere of Banach space $X$ is a nearly rotund point of the unit ball of the bidual space. For any closed convex set $C\subset X$ and $x\in X\setminus C$ with $P_C\left(x\right)\ne \varnothing$, we give a series of characterizations such that $C$ is approximatively compact or approximatively weakly compact for $x$ by using three types of nearly convex points. Furthermore, making use of an S point, we present a characterization such that the convex subset $C$ is approximatively compact for some $x$ in $X\setminus C$. We also establish a relationship between nested sequence of balls and the approximate compactness of the closed convex subset $C$ for some $x\in X\setminus C$.