Generic Gâteaux-differentiability of convex functions on small sets

Generic G&eaux- Differentiability of Convex Functions on Small Sets

The theory of generic differentiability of convex functions on Banach spaces is by now a well-explored part of infinite-dimensional geometry. All the attempts to solve this kind of problem have in common, as a working hypothesis, one special feature of the finite-dimensional case. Namely, convex functions are always considered to be defined on convex sets with nonempty interior. But typically, ...

Generic Fr6chet -differentiability of Convex Functions on Small Sets

Differentiability of convex functions outside small sets

Small sets. We are going to present some results of the following type: under a suitable assumption on a normed space X, each continuous convex function, defined on an open convex set A ⊂ X, is (Gâteaux or Fréchet) differentiable outside a set which is in some sense small. It is natural to ask that a nonempty family S ⊂ 2X whose elements are considered “small sets” satisfy the following conditi...

On Differentiability of Metric Projections onto Moving Convex Sets

We consider properties of the metric projections onto moving convex sets in normed linear spaces. Under certain conditions about the norm, directional diierentiability (of higher order) of the metric projections at boundary points is characterized. The characterization is formulated in terms diierentiability of multifunctions and properties of the set-derivatives are shown.

On Fréchet differentiability of convex functions on Banach spaces

Equivalent conditions for the separability of the range of the subdifferential of a given convex Lipschitz function f defined on a separable Banach space are studied. The conditions are in terms of a majorization of f by a C-smooth function, separability of the boundary for f or an approximation of f by Fréchet smooth convex functions.