Convexity in Banach spaces: some recent results

Abstract

We all recognize that some of the most fruitful ideas in a particular area of mathematics may well have originated elsewhere. In the case of convexity in Banach spaces, this occurred in 1967, when Marc Rieffel wanted to do a thorough classroom presentation of the Radon-Nikodÿm theorem for Banach space-valued measures. In formulating a condition on the range of such a measure which would be sufficient for the validity of the Radon-Nikodým theorem, yet would avoid earlier compactness hypotheses, he introduced the notion of a “dentable” set [41]. This was the start of a remarkable chain of results connecting vector-valued integration, the extremal structure of bounded convex sets and generic Fréchet differentiability of convex continuous functions. The details of this story through 1976 have been related in a lively and thorough manner by J. Diestel and J. Uhl in their monograph [14], while the more recent results are covered in the forthcoming lecture notes by R. Bourgin [9]. We will present a portion of the chain, centering our attention on the extremal structure of bounded closed convex subsets of Banach spaces (and differentiability properties of some closely related convex functions). This first section contains a review of the notions of exposed points, strongly exposed points, dentable sets and the Radon-Nikodým property (RNP). In Section 2 there is a discussion of the differentiability of convex functions on Banach spaces and the duality between Fréchet differentiability and strongly exposed points, followed by one of the main recent results in the subject (Theorem 2.8): A bounded closed convex set with the RNP is the closed convex hull of its strongly exposed points. The concluding Section 3 describes the Krein-Milman property and its relation to the RNP, as well as the duality between Asplund spaces and the RNP.KeywordsBanach SpaceExtreme PointClosed Convex HullAsplund SpaceBounded Closed ConvexThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.