Note on H-convex functions.
On a globalization property
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 if there is a neighbourhood U of such that f(x) - f() ≥ ϕ(x) - ϕ() for all x ∈ U. A function ϕ ∈ Φ is called a global Φ-subgradient of f at 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...
On a property of pseudometrics and uniformities near to convexity
On Carathéodory's and Krein-Milman's theorems in fully ordered groups
On convex metric spaces V
On cyclic α(·)-monotone multifunctions
Let (X,d) be a metric space. Let Φ be a linear family of real-valued functions defined on X. Let be a maximal cyclic α(·)-monotone multifunction with non-empty values. We give a sufficient condition on α(·) and Φ for the following generalization of the Rockafellar theorem to hold. There is a function f on X, weakly Φ-convex with modulus α(·), such that Γ is the weak Φ-subdifferential of f with modulus α(·), .
On differentiability of strongly α(·)-paraconvex functions in non-separable Asplund spaces
In Rolewicz (2002) it was proved that every strongly α(·)-paraconvex function defined on an open convex set in a separable Asplund space is Fréchet differentiable on a residual set. In this paper it is shown that the assumption of separability is not essential.
On generalizations of fuzzy metric spaces
The aim of the paper is to present three-variable generalizations of fuzzy metric spaces in sense of George and Veeramani from functional and topological points of view, respectively. From the viewpoint of functional generalization, we introduce a notion of generalized fuzzy 2-metric spaces, study their topological properties, and point out that it is also a common generalization of both tripled fuzzy metric spaces proposed by Tian et al. and -fuzzy metric spaces proposed by Sedghi and Shobe. Since...
On hyperplanes and semispaces in max–min convex geometry
The concept of separation by hyperplanes and halfspaces is fundamental for convex geometry and its tropical (max-plus) analogue. However, analogous separation results in max-min convex geometry are based on semispaces. This paper answers the question which semispaces are hyperplanes and when it is possible to “classically” separate by hyperplanes in max-min convex geometry.
On Nash theorem
On the algebraic properties of convex bodies and some applications.
On the rank of a topological convexity
On the Rockafellar theorem for -monotone multifunctions
Let X be an arbitrary set, and γ: X × X → ℝ any function. Let Φ be a family of real-valued functions defined on X. Let be a cyclic -monotone multifunction with non-empty values. It is shown that the following generalization of the Rockafellar theorem holds. There is a function f: X → ℝ such that Γ is contained in the -subdifferential of f, .
On α(·)-monotone multifunctions and differentiability of γ-paraconvex functions
Let (X,d) be a metric space. Let Φ be a family of real-valued functions defined on X. Sufficient conditions are given for an α(·)-monotone multifunction to be single-valued and continuous on a weakly angle-small set. As an application it is shown that a γ-paraconvex function defined on an open convex subset of a Banach space having separable dual is Fréchet differentiable on a residual set.
Paraconvex analysis
Pseudo-boundaries and pseudo-interiors for topological convexities [Book]
Representation theorem for convex effect algebras
Effect algebras have important applications in the foundations of quantum mechanics and in fuzzy probability theory. An effect algebra that possesses a convex structure is called a convex effect algebra. Our main result shows that any convex effect algebra admits a representation as a generating initial interval of an ordered linear space. This result is analogous to a classical representation theorem for convex structures due to M.H. Stone.
Saddle point theorems on generalized convex spaces.
Semiconvex compacta
We define and investigate a generalization of the notion of convex compacta. Namely, for semiconvex combination in a semiconvex compactum we allow the existence of non-trivial loops connecting a point with itself. It is proved that any semiconvex compactum contains two non-empty convex compacta, the center and the weak center. The center is the largest compactum such that semiconvex combination induces a convex structure on it. The convex structure on the weak center does not necessarily coincide...
Simplicial homology of preconvexity spaces