On reduced pairs of bounded closed convex sets.
In this paper certain criteria for reduced pairs of bounded closed convex set are presented. Some examples of reduced and not reduced pairs are enclosed.
Pairs of convex bodies in a hyperspace over a Minkowski two-dimensional space joined by a unique metric segment
Let be a Minkowski space with a unit ball and let be the Hausdorff metric induced by in the hyperspace of convex bodies (nonempty, compact, convex subsets of ℝ). R. Schneider [RSP] characterized pairs of elements of which can be joined by unique metric segments with respect to for the Euclidean unit ball Bⁿ. We extend Schneider’s theorem to the hyperspace over any two-dimensional Minkowski space.
Minkowski difference and Sallee elements in an ordered semigroup
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 on p. 2) is a compact convex subset of a topological vector space such that for all subsets the Minkowski difference of the sets and is a summand of . The family of Sallee sets characterizes the Minkowski subtraction, which is important to the arithmetic of compact convex sets (see [5]). Sallee...
On Simons' version of Hahn-Banach-Lagrange theorem
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).
Minimal pairs of bounded closed convex sets as minimal representations of elements of the Minkowski-Rådström-Hörmander spaces
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.
Support functions and subdifferentials
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.
Algebraic Separation and Shadowing of Arbitrary Sets
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 by another compact convex set 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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