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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Erik Alfsen, Compact convex sets and boundary integrals, Ergeb. der Math., vol 57, Springer-Verlag, Berlin, Heidelburg, New York (1971).
Anantharaman, R., T. Lewis and J.H.M. Whitfield, Smoothability and dentability in Banach spaces, Canad. Math. Bull. 24 (1), (1981), 59–68.
Asimow, L., and A.J. Ellis, “Convexity Theory and its Applications in Functional Analysis” London Math. Soc. Monograph No. 16, Academic Press, N.Y. 1980.
Edgar Asplund and R.T. Rockafellar, Gradients of convex functions, Trans. Amer. Math. Soc. 139 (1969), 433–467.
Bishop, E., and R.R. Phelps, The support functionals of a convex set, Proc. Sympos. Pure Math., vol 7, Amer. Math. Soc., Providence, R.I. (1963), 27–35.
Bourgain, J., Dentability and the Bishop-Phelps property, Israel J. Math. 28 (1977), 265–271.
Bourgain, J., A geometric characterization of the Radon-Nikodÿm property in Banach spaces, Compositio Math. 36 (1978), 3–6.
Bourgain, J., and M. Talagrand, Dans un espace de Banach reticulé solide, la propriété de Radon- Nikodÿm et celle de Krein-Milman sont equivalents, Proc. Amer. Math. Soc.; 81 (1981), 93–96.
Bourgin, R., Geometric aspects of convex sets with the Râdon-Nikodÿtn property, Springer Lecture Notes (to appear).
Bourgin, R., and G.A. Edgar, Noncompact simplexes in Banach spaces with the Radon- Nikodÿm property, J. Funct. Anal. 23 (1976), 162–176.
Collier, J.B., The dual of a space with the Radon-Nikodÿin property, Pacific J. Math. 64 (1976), 103–106.
Davis, W.J., and R.R. Phelps, The Radon-Nikod m property and dentable sets in Banach spaces, Proc. Amer. Math. Soc. 45 (1974), 119–122.
Joseph Diestel, Geometry of Banach Spaces-Selected Topics, Lecture Notes in Math. 485, Springer-Verlag, Berlin-Heidelberg-New York, (1975).
Diestel, J., and J.J. Uhl, Jr., Vector Measures, Math. Surveys Nr. 15, Amer. Math. Soc., Providence, R. I. (1977).
Edgar, G.A., A noncompact Choquet theorem, Proc. Amer. Math. Soc. 49 (1975), 354–358.
Edgar, G.A., Extremal integral representations, J. Functional Anal. 23 (1976), 145–161.
Ivar Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. (N.S.) 1, (1979), 443–474.
Fitzpatrick, S.P., Monotone operators and dentability, Bull. Australian Math. Soc. 18 (1978), 77–82.
Huff, R.E., Dentability and the Radon-Nikodÿm property, Duke Math. J. 41 (1974), 111–114.
Huff, R.E., and P.D. Morris, Dual spaces with the Krein-Milman property have the Radon- Nikodÿm property, Proc. Amer. Math. Soc. 49 (1975), 104–108.
Huff, R.E., and P.D. Morris, Geometric characterizations of the Radon-Nikodÿm property in Banach spaces, Studia Math. 56 (1976), 157–164.
R. Israel, Convexity in the Theory of Lattice Gases, Princeton Series in Physics, Princeton Univ. Press, Princeton, N.J. (1979).
James, R.C., Weakly compact sets, Trans. Amer. Math. Soc. 113 (1964), 129–140.
James, R.C., Reflexivity and the sup of linear functionals, Israel J. Math. 13 (1972), 289–300.
James, R.C., Subbushes and extreme points in Banach spaces, Abstracts Amer. Math. Soc. 2 (1981), 599.
Jayne, J., and C.A. Rogers, The extremal structure of convex sets, J. Functional Anal. 26 (1977), 251–288.
Kenderov, P.S., Monotone operators in Asplund spaces, C.R. Acad. Bulgare Sci. 30 (1977), 963–964.
Klee, V.L., Jr., Extremal structure of convex sets, Archiv der Math. 8 (1957), 234–240.
Kunen, K., and H. Rosenthal, Martingale proofs of some geometrical results in Banach theory, preprint, 33pp.
Larman, D.G., and R.R. Phelps, Gateaux differentiability of convex functions on Banach spaces, J. Lond. Math. Soc. (2) 20 (1979), 115–127.
Maynard, H.B., A geometrical characterization of Banach spaces having the Radon- Nikodÿm property, Trans. Amer. Math. Soc. 185 (1973), 493–500.
Morris, P.D., Characterizations of closed convex sets with the RNP, due to J. Bourgain, John Rainwater Seminar Notes, Univ. of Wash., Jan. 1977.
Isaac Namioka, Characterizations of closed convex sets with the RNP, following Bourgain and Stegall, Rainwater Seminar, Univ. of Wash., Oct. 1976, 11 pp.
Isaac Namioka, and R.R. Phelps, Banach spaces which are Asplund spaces, Duke Math. J. 42 (1975), 735–750.
Phelps, R.R., A representation theorem for bounded convex sets, Proc. Amer. Math. Soc. 11 (1960), 976–983.
Phelps, R.R., Lectures on Choquet’s theorem, D. Van Nostrand, Princeton, N.J. (1966).
Phelps, R.R., Support cones in Banach spaces and their applications, Advances Math. 13 (1974), 1–19.
Phelps, R.R., Dentability and extreme points in Banach spaces, J. Funct. Anal. 16 (1974), 78–90.
Phelps, R.R., Differentiability of convex functions on Banach spaces, Lecture notes (unpublished), University College London, January, 1978, 65 pp.
Phelps, R.R., Integral representation for elements of convex sets, Studies in Functional Analysis, MAA Studies in Math., vol. 21, R.G. Bartle, Ed., Math. Assoc. Amer. (1980), 115–157.
Rieffel, M.A., Dentable subsets of Banach spaces, with application to a Radon-Nikodÿm theorem, Functional Analysis (Proc. Conf., Irvine, Calif. (1966), B.R. Gelbaum, Ed., Academic Press, London, (1967), 71–77.
Wayne, A., Roberts and Dale E. Varberg, Convex Functions, Academic Press, New York- London (1973).
Jean Saint-Raymond, Représentation intégrale dans certains convexes, Sém. Choquet, Initiation à l’Analyse, 14e année, 1974/75, 1 1 pp.
Smul’yan, V.L. Sur la structure de la sphère unitaire dans l’espace de Banach, Mat. Sbomik (N.S.) 9, (51) (1941), 545–561.
Charles Stegall, The Radon-Nikodÿm property in conjugate Banach spaces, Trans. Amer. Math. Soc. 206 (1975), 213–223.
Charles Stegall, The duality between Asplund spaces and spaces with the Radon-Nikodÿm property, Israel J. Math. 29 (1978), 408–412.
Charles Stegall, Optimization of functions on certain subsets of Banach spaces, Math. Annalen 236 (1978), 171–176.
Francis Sullivan, On the duality between Asplund spaces and spaces with the Radon- Nikodÿm property, Proc. Amer. Math. Soc. 71 (1978), 155–156.
Thomas, E.G.F., A converse to Edgar’s theorem, Measure theory, Oberwolfach 1979 (Proc. Conf., Oberwolfach, 1979) Lecture Notes in Math. 794, Springer-Verlag, Berlin (1980), 497512.
Bourgain, J., La propriété de Radon-Nikodÿm, Publ. Math. de l’Université Pierre et Marie Curie, No. 36, (1979), 104.
Choquet, G., Lectures on Analysis, vol. II, J. Marsden, T. Lance, S. Gelbart (eds.), W.A. Benjamin (1969).
Giles, J.R., Convex analysis with application in differentiation of convex functions, Res. Notes in Math. 58, Pitman, Boston-London-Melbourne, 1982.
Weizsäcker, H.v., and G. Winkler, Non-compact extremal integral representations: Some probabilistic aspects, Functional Analysis: Surveys and Recent Results II, K.-D. Bierstedt, B. Fuchssteiner (eds.), North-Holland, (1980), 115–148.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1983 Springer Basel AG
About this chapter
Cite this chapter
Phelps, R.R. (1983). Convexity in Banach spaces: some recent results. In: Gruber, P.M., Wills, J.M. (eds) Convexity and Its Applications. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5858-8_12
Download citation
DOI: https://doi.org/10.1007/978-3-0348-5858-8_12
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-5860-1
Online ISBN: 978-3-0348-5858-8
eBook Packages: Springer Book Archive