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In the theory of optimization an essential role is played by the differentiability of convex functions. In this paper we shall try to extend the results concerning differentiability to a larger class of functions called strongly α(·)-paraconvex. Let (X, .) be a real Banach space. Let f (x) be a real valued strongly α(·)-paraconvex function defined on an(More)
In 1933 S. Mazur [4] proved the following Theorem 1. Let (X, ·) be a separable real Banach space. Let f be a real-valued convex continuous function defined on an open convex subset Ω ⊂ X. Then there is a subset A ⊂ Ω of the first category such that f is Gateaux differentiable on Ω \ A. The result of Mazur was a starting point for the theory of(More)
Let (X, ·) be a real Banach space. Let C be a closed convex set in X. By a drop D(x, C) determined by a point x ∈ X, x / ∈ C, we shall mean the convex hull of the set {x} ∪ C. We say that C has the drop property if C = X and if for every nonvoid closed set A disjoint with C, there exists a point a ∈ A such that D(a, C) ∩ A = {a}. For a given C a sequence {x(More)
It is shown that in metric spaces each (α, φ)-meagre set A is uniformly very porous and its index of uniform v-porosity is not smaller than k−α 3k+α , provided that φ is a strictly k-monotone family of Lipschitz functions and α < k. The paper contains also conditions implying that a k-monotone family of Lipschitz functions is strictly k-monotone. Let (X, d)(More)
The notion of nearly uniformly convex Banach spaces was introduced by Huff [2] and independently by Goebel and Sekowski [1]. In [7] it was shown that in a certain way it is a uniformization of drop property for norms. In [3] the authors considered drop property for convex sets. In the present paper they investigate nearly uniformly convex sets which need(More)
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