Gâteaux differentiability of cone-monotone and pointwise Lipschitz functions
The research presented concerns almost everywhere differentiability of cone monotone and Lipschitz mappings. One can consider as a starting point for these questions the standard Rademacher theorem about differentiability of Lipschitz mappings between finite-dimensional spaces. The present paper gives a geometric description of a sigma-ideal of exceptional subsets of a separable Banach space with respect to which cone-monotone real-valued functions are Gâteaux differentiable almost everywhere. It is proved that in this case the result is the best possible. The sigma-ideal described in the paper is contained in the sigma-ideal \(\tilde C\) previously described in [\textit{D. Preiss} and \textit{L. Zajíček}, Isr. J. Math. 125, 1--27 (2001; Zbl 1022.46026)]. The two sigma-ideals coincide in the finite-dimensional case, and it is still not known whether they are the same in the infinite-dimensional case. It is also shown in the paper that the same sigma-ideal described in the present paper works for generalisations of Rademacher's and Stepanov's theorems for metric and weak\(^*\) differentiability and for Gâteaux and Hadamard differentiability of mappings with values in spaces with the Radon-Nikodým property.
scientific article; zbMATH DE number 6375693
scientific article