Convexities and Existence of the Farthest Point

Z H Zhang,C Y Liu

doi:10.1155/2011/139597

Abstract

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.

Highlights

Let X be a Banach space, and let X∗ be its dual space
It is known that LUR, WLUR, midpoint locally uniform rotundity MLUR and weakly midpoint locally uniform rotundity WMLUR are four important convexities in the geometric theory of Banach spaces
We prove that FB x is single valued for all x ∈ Q

Summary

Introduction

Let X be a Banach space, and let X∗ be its dual space. Let us denote by B X and S X the closed unit ball and the unit sphere of X, respectively. It is known that LUR, WLUR, midpoint locally uniform rotundity MLUR and weakly midpoint locally uniform rotundity WMLUR are four important convexities in the geometric theory of Banach spaces. We prove a sufficient condition that the farthest point map FB is single valued in a residual subset of X when X is very convex

Some Counterexamples about Convexities

Convexities and Existence of the Farthest Point