Generic Fréchet differentiability of convex functions dominated by a lower semicontinuous convex function. (English) Zbl 0937.46039
It is shown that a lower semi-continuous convex function \(f:E\to\mathbb R\cup\{+\infty\}\) with \(f^{-1}(\mathbb R)\neq\emptyset\) has the property that every such function \(g\) with \(g\leq f\) is Fréchet differentiable on a dense \(G_\delta\)-subset of the interior of \(g^{-1}(\mathbb R)\) if and only if the \(w^*\)-closed convex hull of the image of the subdifferential map of \(f\) has the Radon-Nikodým property. The proof uses and extends a (so far) unpublished theorem in [the first named authors, “Generic Fréchet differentiability of convex functions on non-Asplund spaces”].
Reviewer: A.Kriegl (Wien)
MSC:
| 46G05 | Derivatives of functions in infinite-dimensional spaces |
| 46B50 | Compactness in Banach (or normed) spaces |
Keywords:
Asplund space; generic Fréchet differentiability; lower semi-continuous convex functions; Radon-Nikodým propertyReferences:
| [1] | Asplund, E., Fréchet differentiability of convex functions, Acta Math., 121, 31-47 (1968) · Zbl 0162.17501 |
| [2] | Bishop, E.; Phelps, R. R., A proof that every Banach space is subreflexive, Bull. Amer. Math. Soc., 67, 97-98 (1961) · Zbl 0098.07905 |
| [3] | Bourgin, R. D., Geometric aspect of convex sets with the Radon-Nikodým property, Lecture Notes in Math., 993 (1983) · Zbl 0512.46017 |
| [4] | Brøndsted, A.; Rockafellar, R. T., On the subdifferentiability of convex functions, Proc. Amer. Math. Soc., 16, 605-611 (1965) · Zbl 0141.11801 |
| [5] | Cheng, Lixin, Shi, Shuzhong, Generic Fréchet differentiability of convex functions on non-Asplund spaces; Cheng, Lixin, Shi, Shuzhong, Generic Fréchet differentiability of convex functions on non-Asplund spaces · Zbl 0909.46013 |
| [6] | Giles, J. R.; Sciffer, S., Separable determination of Fréchet differentiability of convex functions, Bull. Austral. Math. Soc., 52, 161-167 (1995) · Zbl 0839.46036 |
| [7] | Kenderov, P. S., Monotone operator in Asplund spaces, C. R. Acad. Bulgare Sci., 30, 963-964 (1977) · Zbl 0377.47036 |
| [8] | Namioka, I.; Phelps, R. R., Banach spaces which are Asplund spaces, Duke Math. J., 42, 735-750 (1975) · Zbl 0332.46013 |
| [9] | Phelps, R. R., Convex functions, differentiability and monotone operators, Lecture Notes in Math., 1364 (1989) · Zbl 0658.46035 |
| [10] | Preiss, D., Differentiability of Lipschitz functions on Banach spaces, J. Funct. Anal., 91, 312-345 (1990) · Zbl 0711.46036 |
| [11] | Preiss, D., Differentiability and measures in Banach spaces, Proceedings of the International Congress on Mathematics (1990), p. 923-929 · Zbl 0762.46029 |
| [12] | Stegall, C., The duality between Asplund spaces and spaces with the Radon-Nikodým property, Israel J. Math., 29, 408-412 (1978) · Zbl 0374.46015 |
| [13] | Wee-Kee, Tang, On Fréchet differentiability of convex functions on Banach spaces, Comment. Math. Univ. Carolin., 36, 249-253 (1995) · Zbl 0831.46045 |
| [14] | Congxin, Wu; Lixin, Cheng, Differentiability of norms on \(c_0l^∞\), Northeast. Math. J., 12, 153-160 (1996) · Zbl 0861.46012 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
