Yet more on the differentiability of convex functions

Author:
John Rainwater

Journal:
Proc. Amer. Math. Soc. **103** (1988), 773-778

MSC:
Primary 46G05; Secondary 26B05, 46B22, 58C06, 90C48

DOI:
https://doi.org/10.1090/S0002-9939-1988-0947656-7

MathSciNet review:
947656

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Abstract | References | Similar Articles | Additional Information

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).

**[1]**Edgar Asplund,*Fréchet differentiability of convex functions*, Acta Math.**121**(1968), 31-47. MR**0231199 (37:6754)****[2]**J. P. R. Christensen,*Theorems of Namioka and R. E. Johnson type for upper semicontinuous and compact-valued set-valued mappings*, Proc. Amer. Math. Soc.**86**(1982), 649-655. MR**674099 (83k:54014)****[3]**Joseph Diestel,*Geometry of Banach spaces -- selected topics*, Lecture Notes in Math., vol. 485, Springer-Verlag, 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*, Springer-Verlag, 1975. MR**0410335 (53:14085)****[6]**P. S. Kenderov,*Monotone operators in Asplund spaces*, C. R. Acad. Bulgare Sci.**30**(1977), 963-964. MR**0463981 (57:3919)****[7]**I. Namioka,*Separate continuity and joint continuity*, Pacific J. Math.**51**(1974), 515-531. MR**0370466 (51:6693)****[8]**Isaac Namioka and R. R. Phelps,*Banach spaces which are Asplund spaces*, Duke Math. J.**42**(1975), 735-749. MR**0390721 (52:11544)****[9]**R. R. Phelps,*Some topological properties of support points of convex sets*, Israel J. Math.**13**(1972), 327-336. 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, Springer-Verlag, 1985, pp. 158-168. MR**827772 (87k:46094)****[12]**M. E. Verona,*More on the differentiability of convex functions*, Proc. Amer. Math. Soc.**103**(1988), 137-140. MR**938657 (89f:58016)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1988-0947656-7

Keywords:
Convex functions,
convex sets,
support points,
Asplund spaces,
weak Asplund spaces,
usco maps

Article copyright:
© Copyright 1988
American Mathematical Society