Generic Gâteaux-differentiability of convex functions on small sets (original) (raw)

Gateaux Differentiability Revisited

Applied Mathematics & Optimization, 2021

We revisit some basic concepts and ideas of the classical differential calculus and convex analysis extending them to a broader frame. We reformulate and generalize the notion of Gateaux differentiability and propose new notions of generalized derivative and generalized subdifferential in an arbitrary topological vector space. Meaningful examples preserving the key properties of the original notion of derivative are provided.

Separable determination of Fréchet differentiability of convex functions

Bulletin of the Australian Mathematical Society, 1995

For a continuous convex function on an open convex subset of any Banach space a separability condition on its image under the subdifferential mapping is sufficient to guarantee the generic Frechet differentiability of the function. This gives a direct insight into the characterisation of Asplund spaces.

Generic Gateaux differentiability via smooth perturbations

Bulletin of the Australian Mathematical Society, 1997

We prove that in a Banach space with a Lipschitz uniformly Gateaux smooth bump function, every continuous function which is directionally differentiable on a dense Gj subset of the space, is Gateaux differentiable on a dense Gs subset of the space. Applications of this result are given. The usual applications of variational principles in Banach spaces are to differentiability of real valued functions. For example, the papers [1] and [2] contain results about Gateaux differentiability on dense sets. An application of Ekeland's variational principle to generic Frechet differentiability is given in the proof of the famous Ekeland-Lebourg theorem (see [4]). In [6] an application of the smooth variational principle to generic Gateaux differentiability is presented. In this paper we prove some results about generic Gateaux differentiability of directionally differentiable functions. The tool for proving the main result (Theorem 2) is Proposition 1, which localises precisely the d-minimum point of the perturbed function. The estimate of this localisation is the same as in the Ekeland variational principle. Denote by Lj the Lipschitz constant of a Lipschitz function / : E-» R and by 5, B[x;r] (respectively B(x;r))-the unit sphere of E and the closed (respectively open) ball with center x and radius r. A function b : E-> R is said to be a bump function, if there exists a bounded subset supp b, such that b(x)-0 for every x 0 supp b.

On weak Hadamard differentiability of convex functions on Banach spaces

Bulletin of the Australian Mathematical Society, 1996

We study two variants of weak Hadamard differentiability of continuous convex functions on a Banach space, uniform weak Hadamard differentiability and weak Hadamard directional differentiability, and determine their special properties on Banach spaces which do not contain a subspace topologically isomorphic to l\.

Differentiability of cone-monotone functions on separable Banach space

Proceedings of the American Mathematical Society, 2004

Abstract: Motivated by applications to (directionally) Lipschitz functions, we provide a general result on the almost everywhere Gâteaux differentiability of real-valued functions on separable Banach spaces, when the function is monotone with respect to an ordering induced by a convex cone with non-empty interior. This seemingly arduous restriction is useful, since it covers the case of directionally Lipschitz functions, and necessary. We show by way of example that most results fail more generally.