ABSTRACT
The geometrical approach to the study of Banach spaces is connected with the unit ball of the space, i.e., the set BX(0,1) = {x : ‖x‖ < 1}. Since Banach spaces have many topological properties that are useful in applications, the following problem is quite natural: What properties of a Banach space are expressible in isometric terms? The earliest considerations of certain aspects of this problem are in the work of J. A. Clarkson and M. G. Krein. They introduced (independently) the very important class of strictly convex spaces (also called rotund spaces) and gave several interesting properties of this class of Banach spaces. Also, Clarkson considered the class of uniformly convex spaces, which seems to be the most extensively studied class of Banach spaces from the above-mentioned point of view. Since the work of Clarkson and Krein the study of classes of spaces related to the above has attracted the attention of a great number of mathematicians. One reason for this is the fact that certain theorems of classical analysis as well as of modern analysis can be proved in greater generality only in such spaces. It is to be noted that these classes of Banach spaces have proved useful in many areas of mathematics, such as probability theory, function theory, harmonic analysis, fixed point theory, and operator theory. In what follows we describe these classes of Banach spaces as well as some applications.
