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Generic Fréchet differentiability of convex functions on non-Asplund spaces. (English) Zbl 0905.46029

A Banach space is said to be an Asplund space if every real-valued continuous convex function defined on a non-empty convex open subset of the space is Fréchet differentiable on a dense \(G_\delta\) subset of its domain. It is well-known that the Banach space \(E\) is an Asplund space if and only if its dual \(E^*\) has the Radon-Nikodým property. However, on a non-Asplund space, one can still find many nontrivial convex functions which are Fréchet differentiable on a dense \(G_\delta\) subset of its domain.
Recently W. K. Tang [Commenta. Math. Univ. Carol. 36, No. 2, 249-253 (1995; Zbl 0831.46045)] obtained the following:
Theorem T. Suppose that \(f\) is a convex Lipschitz function on a separable Banach space \(E\). Then the following statements are equivalent:
(i) every continuous convex function \(g\) on \(E\) with \(g\leq f\) is Fréchet differentiable on a dense \(G_\delta\) subset of \(E\);
(ii) the image of the subdifferential map \(\partial f\) of \(f\), \(\partial f(E)= \{x^*\in E^*: x^*\in\partial f(x)\), \(x\in E\}\) is separable.
Also, J. R. Giles and S. Sciffer [Bull. Aust. Math. Soc. 52, No. 1, 161-167 (1995; Zbl 0839.46036)] showed the following:
Theorem GS. Let \(f\) be a continuous convex function on a Banach space \(E\). If the image of the subdifferential map \(\partial f\) of \(f\) is separable on each separable subspace of \(E\), then \(f\) is Fréchet differentiable on a dense \(G_\delta\) subset of \(E\).
Both theorems show that on a separable Banach space \(E\), there exist many generically Fréchet differentiable continuous convex functions, even if \(E\) is not necessarily an Asplund space. The authors of this note have the following final theorem.
Theorem. Let \(f\) be a proper lower semicontinuous (l.s.c.) convex function on a Banach space \(E\) and its effective domain \(\text{dom }f\) be open. Then the following statements are equivalent:
(i) every proper l.s.c. convex function \(g\) on \(E\) with \(g\leq f\) is Fréchet differentiable on a dense \(G_\delta\) subset of its effective domain \(\text{dom }g\);
(ii) the image of the subdifferential map \(\partial f\) of \(f\), \(\partial f(E)= \{x^*\in E^*: x^*\in\partial f(x)\), \(x\in E\}\), is separable on each separable subspace of \(E\);
(iii) the image of the subdifferential map \(\partial f\) of \(f\), \(\partial f(E)\), has the Radon-Nikodým property;
(iv) the \(w^*\)-closed convex hull of the image of the subdifferential map \(\partial f\) of \(f\), \(w^*\)-\(\text{cl co}[\partial f(E)]\), has the Radon-Nikodým property.
Reviewer: S.Koshi (Sapporo)

MSC:

46G05 Derivatives of functions in infinite-dimensional spaces
46B22 Radon-Nikodým, Kreĭn-Milman and related properties
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References:

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