Some examples concerning rotundity in Banach spaces

Abstract

A Banach space is rotund if the midpoint of each chord of the unit ball lies beneath the surface. In 1936 Clarkson [3] introduced the stronger notion of uniform rotundity. A Banach space is uniformly rotund if the midpoints of all chords of the unit ball whose lengths are bounded below by a positive number are uniformly buried beneath the surface. Since Clarkson's paper many authors have defined and studied geometric properties whose strengths lie between rotundity and uniform rotundity (see [1, 7-9,11, and 12]). Most of these properties can be classified as either localizations or directionalizations of uniform rotundity. The localizations--locally uniformly rotund and midpoint locally uniformly rotund-and the directionalizations--weakly uniformly rotund and uniformly rotund in every direction--have been of particular interest in the literature (see [4, 6, and 13]). In this paper six examples of Banach spaces are given that illustrate the distinctions among these generalizations of uniform rotundity as well as the independence of the localizations and directionalizations.