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Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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How to define "convex functions" on differentiable manifolds

100%
Rolewicz S.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2009
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tom 29
|
nr 1
7-17
EN
In the paper a class of families 𝓕(M) of functions defined on differentiable manifolds M with the following properties: $1_{𝓕}$. if M is a linear manifold, then 𝓕(M) contains convex functions, $2_{𝓕}$. 𝓕(·) is invariant under diffeomorphisms, $3_{𝓕}$. each f ∈ 𝓕(M) is differentiable on a dense $G_{δ}$-set, is investigated.
2
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On a globalization property

100%
Rolewicz S.
Applicationes Mathematicae
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1993-1995
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tom 22
|
nr 1
69-73
EN
Let (X,τ) be a topological space. Let Φ be a class of real-valued functions defined on X. A function ϕ ∈ Φ is called a local Φ-subgradient of a function f:X → ℝ at a point $x_0$ if there is a neighbourhood U of $x_0$ such that f(x) - f($x_0$) ≥ ϕ(x) - ϕ($x_0$) for all x ∈ U. A function ϕ ∈ Φ is called a global Φ-subgradient of f at $x_0$ if the inequality holds for all x ∈ X. The following properties of the class Φ are investigated: (a) when the existence of a local Φ-subgradient of a function f at each point implies the existence of a global Φ-subgradient of f at each point (globalization property), (b) when each local Φ-subgradient can be extended to a global Φ-subgradient (strong globalization property).
3
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On /()X\)-convex functions

100%
Rolewicz S.
Commentationes Mathematicae
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2013
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tom 53
|
nr 2
EN
Let \(X\) be a Banach space. Let \(f(\cdot)\) be a real valued function defined on an open convex set \(\Omega \subset X^*\), where \(X^*\) as usual denote the conjugate space. We say that the function \(f(\cdot)\) is \(X$\)convex, if there is a set \(\Phi_f \subset X\) such that $$ f(x^*)= sup_{x \in \Phi_f, r \in \R} x^*(x)+r. \eqno{(1)}$$ In the paper it will be shown that if \(X\) is separable, then the function \(f(\cdot)\) is Frechet differentiable on a dense \(G_{\delta}\) set.
4
Content available remote

Coefficient of orthogonal convexity of some Banach function spaces

64%
Kolwicz P., Rolewicz S.
Studia Mathematica
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2004
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tom 164
|
nr 2
121-138
EN
We study orthogonal uniform convexity, a geometric property connected with property (β) of Rolewicz, P-convexity of Kottman, and the fixed point property (see [19, [20]). We consider the coefficient of orthogonal convexity in Köthe spaces and Köthe-Bochner spaces.
5
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On σ-porous and Φ-angle-small sets in metric spaces

32%
Rolewicz S.
Control and Cybernetics
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2003
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tom 32
|
nr 3
671-681
6
Content available remote

Paraconvex analysis

23%
Rolewicz S.
Control and Cybernetics
|
2005
|
tom 34
|
nr 3
951-965
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