Differentiability of convex functions outside small sets
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چکیده
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 conditions: (a) A ∈ S, B ⊂ A ⇒ B ∈ S; (b) An ∈ S for each n ∈ N ⇒ ⋃ n∈NAn ∈ S; (c) A ∈ S ⇒ A + v ∈ S for each v ∈ X (that is, S is translation invariant); (d) S contains no nonempty open set. Notice that (a) and the assumption that S is nonempty imply that ∅ ∈ S. Each family satisfying (a) and (b) is called a σ-ideal. Thus a family of small sets has to be a nonempty translation invariant σ-ideal that cointains no open ball (of positive radius).
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