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- Stefan Rolewicz
- 2005

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)

- Stefan Rolewicz
- 2007

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 6= 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… (More)

In the paper a class of families F(M) of functions defined on differentiable manifolds M with the following properties: 1F . if M is a linear manifold, then F(M) contains convex functions, 2F . F(·) is invariant under diffeomorphisms, 3F . each f ∈ F(M) is differentiable on a dense Gδ-set, is investigated.

- S. M. F, Stefan Rolewicz, Stefan Rolewicz
- 2017

© Mémoires de la S. M. F., 1972, tous droits réservés. L’accès aux archives de la revue « Mémoires de la S. M. F. » (http:// smf.emath.fr/Publications/Memoires/Presentation.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une… (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)

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.

- Szczepan Perz, Stefan Rolewicz
- ZOR - Meth. & Mod. of OR
- 1990

The weak Berge hypothesis states that a graph is perfect if and only if its complement is perfect. Previous proofs of this hypothesis have used combinatorial or polyhedral methods. In this paper, the concept of norms related to graphs is used to provide an alternative proof for the weak Berge hypothesis. Zusammenfassung: Die schwache Berge Vermutung sagt… (More)

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