Convexity in Banach spaces: some recent results

摘要: We all recognize that some of the most fruitful ideas in a particular area mathematics may well have originated elsewhere. In case convexity Banach spaces, this occurred 1967, when Marc Rieffel wanted to do thorough classroom presentation Radon-Nikodym theorem for space-valued measures. formulating condition on range such measure which would be sufficient validity Radon-Nikodým theorem, yet avoid earlier compactness hypotheses, he introduced notion “dentable” set [41]. This was start remarkable chain results connecting vector-valued integration, extremal structure bounded convex sets and generic Frechet differentiability continuous functions. The details story through 1976 been related lively manner by J. Diestel Uhl their monograph [14], while more recent are covered forthcoming lecture notes R. Bourgin [9]. will present portion chain, centering our attention closed subsets spaces (and properties closely functions). first section contains review notions exposed points, strongly dentable property (RNP). Section 2 there is discussion functions duality between followed one main subject (Theorem 2.8): A with RNP hull its points. concluding 3 describes Krein-Milman relation RNP, as Asplund RNP.