Convergence of Slices, Geometric Aspects in Banach Spaces and Proximinality

Abstract

Some geometric properties of Banach spaces and proximinality properties in best approximation theory are characterized in terms of convergence of slices. The paper begins with some basic geometric properties of Banach spaces involving slices and their geometric interpretations. Two notions of convergence of sequence sets, called Vietoris convergence and Hausdorff convergence, with their characterizations are presented. It is observed that geometric properties such as uniform convexity, strong convexity, Radon-Riesz property, and strong subdifferentiability of the norm can be characterized in terms of the convergence of slices with respect to the notions mentioned above. Proximinality properties such as approximative compactness and strong proximinality of closed convex subsets of a Banach space are also characterized in terms of convergence of slices.