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Pairs of convex bodies in a hyperspace over a Minkowski two-dimensional space joined by a unique metric segment

100%
Bogdewicz A., Grzybowski J.
Banach Center Publications
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2009
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tom 84
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nr 1
75-88
EN
Let $(ℝ,||·||_𝔹)$ be a Minkowski space with a unit ball 𝔹 and let $ϱ_H^{𝔹}$ be the Hausdorff metric induced by $||·||_{𝔹}$ in the hyperspace 𝒦 of convex bodies (nonempty, compact, convex subsets of ℝ). R. Schneider [RSP] characterized pairs of elements of 𝒦 which can be joined by unique metric segments with respect to $ϱ_H^{Bⁿ}$ for the Euclidean unit ball Bⁿ. We extend Schneider's theorem to the hyperspace $(𝒦²,ϱ_H^{𝔹})$ over any two-dimensional Minkowski space.
2
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Minkowski difference and Sallee elements in an ordered semigroup

100%
Borowska D., Grzybowski J.
Commentationes Mathematicae
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2007
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tom 47
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nr 1
EN
In the manner of Pallaschke and Urbański ([5], chapter 3) we generalize the notions of the Minkowski difference and Sallee sets to a semigroup. Sallee set (see [7], definition of the family \(S\) on p. 2) is a compact convex subset \(A\) of a topological vector space \(X\) such that for all subsets \(B\) the Minkowski difference \(A -B\) of the sets \(A\) and \(B\) is a summand of \(A\). The family of Sallee sets characterizes the Minkowski subtraction, which is important to the arithmetic of compact convex sets (see [5]). Sallee polytopes are related to monotypic polytopes (see [4]). We generalize properties of Minkowski difference and Sallee sets to semigroup and investigate the families of Sallee elements in several specific semigroups.
3
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On Simons' version of Hahn-Banach-Lagrange theorem

81%
Grzybowski J., Przybycień H., Urbański R.
Banach Center Publications
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2014
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tom 102
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nr 1
99-104
EN
In this paper we generalize in Theorem 12 some version of Hahn-Banach Theorem which was obtained by Simons. We also present short proofs of Mazur and Mazur-Orlicz Theorem (Theorems 2 and 3).
4
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Minimal pairs of bounded closed convex sets as minimal representations of elements of the Minkowski-Rådström-Hörmander spaces

81%
Grzybowski J., Pallaschke D., Urbański R.
Banach Center Publications
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2009
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tom 84
|
nr 1
31-55
EN
The theory of minimal pairs of bounded closed convex sets was treated extensively in the book authored by D. Pallaschke and R. Urbański, Pairs of Compact Convex Sets, Fractional Arithmetic with Convex Sets. In the present paper we summarize the known results, generalize some of them and add new ones.
5
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Support functions and subdifferentials

81%
Grzybowski J., Pallaschke D., Urbański R.
Commentationes Mathematicae
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2016
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tom 56
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nr 1
EN
In this paper we study Minkowski duality, i.e. the correspondence between sublinear functions and closed convex sets in the context of dual pairs of vector spaces.
6

Algebraic Separation and Shadowing of Arbitrary Sets

81%
Grzybowski J., Pallaschke D., Urbański R.
Commentationes Mathematicae
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2013
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tom 53
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nr 2
EN
In this paper we consider a generalization of the separation technique proposed in [10,4,7] for the separation of finitely many compact convex sets \(A_i,~i \in I \) by another compact convex set \(S\) in a locally convex vector space to arbitrary sets in real vector spaces. Then we investigate the notation of shadowing set which is a generalization of the notion of separating set and construct separating sets by means of a generalized Demyanov-difference in locally convex vector spaces.
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