## 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

### Citations:

Zbl 0831.46045; Zbl 0839.46036
Full Text:

### References:

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