Inequalities in Uniformly Convex Spaces

Among all Banach spaces, Hilbert spaces are generally regarded as the ones with the simplest geometric structures. The reason for this observation is that certain geometric properties which characterize Hilbert spaces (e.g., the existence of inner product; the para l lelogram law or equivalently the polariza¬tion identity; and the fact that the proximity map or nearest point mapping in Hilbert spaces is Lipschitz with constant 1) make certain problems posed in Hilbert spaces comparatively straightforward and relatively easy to solve. In several applications, however, many problems fall naturally in Banach spaces more general than Hilbert spaces. Therefore, to extend the techniques of so¬lutions of problems in Hilbert spaces to more general Banach spaces, one needs to establish identities or inequalities in general Banach spaces analo¬gous to the ones in Hilbert spaces. As shown by recent works, several authors have conducted worthwhile research in this direction (e.g., Al'ber ([3], [4], [9], Beauzamy [26], Bynum [61, 62], Clarkson [191], Lindenstrauss ([309], [310]), Hanner [247], Kay [276], Lim [306, 303], Lindenstrauss and Tzafriri [311], Prus and Smarzewski [387], Reich [408], Tribunov [491], Xu [509], Xu [523], Xu and Roach [525], and a host of other authors). In this chapter (and also in Chapter 5), we shall describe some of the results obtained primarily within the last thirty years or so. Applications of these results to iterative solutions of nonlinear equations in Banach spaces will be given in subsequent chapters.