Characterizations of some rotundity properties in terms of farthest points

TL;DR

This paper characterizes various rotundity properties of Banach spaces using sets of (almost) farthest points from the unit sphere and introduces new remotality properties related to these sets.

Contribution

It provides novel characterizations of rotundity properties in Banach spaces through the concept of almost farthest points and remotality properties.

Findings

Characterizations of rotund, uniformly rotund, and locally uniformly rotund spaces.

Introduction of new remotality properties based on almost farthest points.

Connections between rotundity and proximinality notions.

Abstract

We characterize rotund, uniformly rotund, locally uniformly rotund and compactly locally uniformly rotund spaces in terms of sets of (almost) farthest points from the unit sphere using the generalized diameter. For this we introduce few remotality properties using the sets of almost farthest points. As a consequence, we obtain some characterizations of the aforementioned rotundity properties in terms of existing proximinality notions.