Infinite dimensional convexity. (Q2760181)

The \textit{Handbook of the geometry of Banach spaces} (Volumes 1 and 2) gathers a quite comprehensive survey of the knowledge accumulated on Banach spaces and their applications inside and outside mathematics. This chapter 15, written by three outstanding experts in the field (among them one of the editors of the whole collection), deals with the structure of infinite-dimensional convex sets and, roughly speaking, with what can be obtained about convex sets through topological methods rather than computations or probabilistic techniques. NEWLINENEWLINEThe first section recalls the classical Krein--Milman and Schauder--Tikhonov theorems. Incidentally, this latter result has recently been extended to the non-locally convex case by \textit{R.\ Cauty} [Fundam.\ Math.\ 170, No.~3, 231--246 (2001; Zbl 0983.54045)]. Section 2 displays the classical Choquet representation theory, with its many applications, while section 3 describes simplices and their structure, and in particular the Poulsen simplex and the various places where it appears. For instance, a remarkable connection beween Kazhdan's property (T) and this object is mentioned. Selection theorems and the relations of simplex theory with \(L^1\)-preduals (commonly called Lindenstrauss spaces) are surveyed. NEWLINENEWLINESection 4 is devoted to the Radon--Nikodym property from a geometrical point of view, and its complex version called the analytic Radon--Nikodym property. It should be stressed that this complex version is rather an analogue of the real notion than a simple extension, since spaces such as \(L^1\) behave differently. Support points and boundaries are investigated in Chapter 5, where it is recalled that the complex form of the Bishop--Phelps theorem fails; incidentally, the relations between complex uniform convexity and the complex version of the Bishop--Phelps theorem are not yet well understood. A new proof of Rodé's theorem, which immediately implies the separable case of James' theorem, is provided. It remains an intriguing fact that, to this day, the non-separable case of James' theorem is considerably more difficult than the separable case. Thin sets and their applications to results of Banach--Steinhaus type (but much deeper) are also displayed. NEWLINENEWLINEThe well-developed theory of polyhedral spaces is covered in Section 6. This theory is partly isometric, but nice characterizations of spaces which are isomorphic to polyhedral spaces are also provided: for instance, a separable Banach space is polyhedral if and only if it has an equivalent norm which locally depends upon finitely many coordinates. It seems to be an open problem to know whether one can replace ``norm'' by ``bump function'' in the above equivalence. Finally, Section 7 gathers miscellaneous topics (stable convex sets, tilings, proximinal and antiproximinal sets) and important open problems. Among them, let us single out these two, which concern the separable Hilbert space \(H\): Let \(C\) be a subset of \(H\) such that every point in \(H\) has a unique nearest point in \(C\). Is \(C\) a convex set? Does there exist a convex tiling of \(H\) which is uniformly bounded from below and above? NEWLINENEWLINEIt should be said in conclusion that this survey is fluently written and carries a lot of information with as few technicalities as possible. Any graduate student in analysis should find it illuminating, and experienced researchers will rely on this authoritative work.NEWLINENEWLINEFor the entire collection see [Zbl 0970.46001].