Convex functions with non-Borel set of Gâteaux differentiability points. (English) Zbl 0970.46026

Let \(X\) be a nonseparable Banach space with a fundamental system (for example, let \(X\) be a nonseparable Hilbert space). The authors show that then there is a convex continuous function \(f\) on \(X\) so that the set \(G(f)\) of points where \(f\) is Gâteaux differentiable is not Borel. This contrasts with the situation in separable Banach spaces, where the set of points of Gâteaux differentiability of a convex continuous function is always a dense \(G_{\delta }\) set. The result answers in the negative a question of J. Rainwater [Bull. Aust. Math. Soc. 42, No. 2, 315-322 (1990; 724.46017)]: it provides an example of a Banach space on which every convex continuous function \(f\) is densely Gâteaux differentiable, yet the set \(G(f)\) is not always Borel.
Let \(X\) be a Banach space with a fundamental system of cardinality continuum. The authors, more concretely, construct a convex continuous function \(f\) on \(X\) with any prescribed intersection of \(G(f)\) with a fixed one-dimensional subspace of \(X\). This generalizes a result of M. Talagrand [C. R. Acad. Sci. Paris, Sér. A 288, 461-464 (1979; Zbl 0398.46037)] who exhibited (by a different method) such a function on \(\ell _1(\mathfrak c)\). The paper contains examples of nonseparable Banach spaces with and without fundamental systems.


46G05 Derivatives of functions in infinite-dimensional spaces
46B20 Geometry and structure of normed linear spaces