Yet more on the differentiability of convex functions
Author:
John Rainwater
Journal:
Proc. Amer. Math. Soc. 103 (1988), 773778
MSC:
Primary 46G05; Secondary 26B05, 46B22, 58C06, 90C48
MathSciNet review:
947656
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Abstract: Generic differentiability theorems are obtained for convex functions which are defined and locally Lipschitzian on the convex subset of nonsupport points of a closed convex subset of a Banach space , which is assumed to be either an Asplund space (for Fréchet differentiability) or to be weakly compactly generated (for Gateaux differentiability).
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 Edgar Asplund, Fréchet differentiability of convex functions, Acta Math. 121 (1968), 3147. MR 0231199 (37:6754)
 [2]
 J. P. R. Christensen, Theorems of Namioka and R. E. Johnson type for upper semicontinuous and compactvalued setvalued mappings, Proc. Amer. Math. Soc. 86 (1982), 649655. MR 674099 (83k:54014)
 [3]
 Joseph Diestel, Geometry of Banach spaces  selected topics, Lecture Notes in Math., vol. 485, SpringerVerlag, 1975. MR 0461094 (57:1079)
 [4]
 John Giles, Convex analysis with application to differentiation of convex functions, Research Notes in Math., no. 58, Pitman, Boston, Mass., 1982.
 [5]
 R. B. Holmes, Geometric functional analysis and its applications, SpringerVerlag, 1975. MR 0410335 (53:14085)
 [6]
 P. S. Kenderov, Monotone operators in Asplund spaces, C. R. Acad. Bulgare Sci. 30 (1977), 963964. MR 0463981 (57:3919)
 [7]
 I. Namioka, Separate continuity and joint continuity, Pacific J. Math. 51 (1974), 515531. MR 0370466 (51:6693)
 [8]
 Isaac Namioka and R. R. Phelps, Banach spaces which are Asplund spaces, Duke Math. J. 42 (1975), 735749. MR 0390721 (52:11544)
 [9]
 R. R. Phelps, Some topological properties of support points of convex sets, Israel J. Math. 13 (1972), 327336. MR 0328558 (48:6900)
 [10]
 C. Stegall, A class of topological spaces and differentiation of functions on Banach spaces, Proc. Conf. on Vector Measures and Integral Representations of Operators, Vorlesungen aus dem Fachbereich. Math., Heft 10 (W. Ruess, ed.), Univ. Essen, 1983. MR 730947 (85j:46026)
 [11]
 , More Gateaux differentiability spaces, Proc. Conf. Banach Spaces, Univ. Missouri, 1984 (N. Kalton and E. Saab, eds.), Lecture Notes in Math., vol. 1166, SpringerVerlag, 1985, pp. 158168. MR 827772 (87k:46094)
 [12]
 M. E. Verona, More on the differentiability of convex functions, Proc. Amer. Math. Soc. 103 (1988), 137140. MR 938657 (89f:58016)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198809476567
PII:
S 00029939(1988)09476567
Keywords:
Convex functions,
convex sets,
support points,
Asplund spaces,
weak Asplund spaces,
usco maps
Article copyright:
© Copyright 1988 American Mathematical Society
