Any abstract convex cone S with a uniformity satisfying the law of cancellation can be embedded in a topological vector space S (Urbański, Bull Acad Pol Sci, Sér Sci Math Astron Phys 24: [709][710][711][712][713][714][715] 1976). We introduce a notion of a cone symmetry and decompose in Theorem 2.12 a quotient vector space S into a topological direct sum of its symmetric subspace S s and asymmetric subspace S a . In Theorem 2.19 we prove a similar decomposition for a normed space S. In section 3 we apply decomposition to Minkowski-Rådström-Hörmander (MRH) space with three best known norms and four symmetries. In section 4 we obtain a continuous selection from a MRH space over R 2 to the family of pairs of nonempty compact convex subsets of R 2 .Keywords Quasidifferential · Abstract convex cone · Cone of nonempty bounded closed convex sets · Minkowski-Rådström-Hörmander space · Hausdorff metric · Demyanov metric · Bartels-Pallaschke norm Mathematics Subject Classifications (2010) 52A07 · 22A20 · 46N10 · 26A27

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In this paper we describe a construction of a large class of hyperconvex metric spaces. In particular, this construction contains well-known examples of hyperconvex spaces such as R 2 with the "river" metric or with the radial one.Further, we investigate linear hyperconvex spaces with extremal points of their unit balls. We prove that only in the case of a plane (and obviously a line) is there a strict connection between the number of extremal points of the unit ball and the hyperconvexity of the space.Some additional properties concerning the notion of hyperconvexity are also investigated.

In this paper properties of the Minkowski-Pontryagin subtraction of closed bounded convex sets are investigated (see Propositions 1-3) and four criteria for summands of closed bounded convex sets are given (see Theorems 1-4).

In this note we generalize the fact that in topological vector spaces the algebraic sum of closed set A and compact set B is closed. We also prove some conditions that are equivalent to reflexivity of Banach spaces.2010 Mathematics Subject Classification. 46A22. 46N10 Keywords. Closed convex sets. InroductionLet X be a Hausdorff topological vector space. By C(X) we denote the family of all closed subsets of X and by K(X) the family of all compact subset of X. For a nonempty subsets A, B ⊂ X we define the algebraic sum (Minkowski sum ) as followIt is wll known that if A, B are closed sets then A + B need not to be closed, but also it is known that if A ∈ C(X) and B ∈ K(X) then A + B ∈ C(X). In this note we prove the last result with some more abstract point of view. Reflexive spacesIn this section we give some results which show the connections of closedness of algebraic sum of sets with reflexivity of Banach spaces.Theorem 2.1. Let X be a Banach space and let B be the closed unit ball in X. Then X is a reflexive Banach space if and only if for every closed convex and bounded subset A of X the algebraic sum A + B is closed.Proof. (Necessity.) Assume that X is a reflexive Banach space, then every closed bounded and convex set is a weakly compact. Since algebraic sum of two weakly compact sets is again weakly compact we conclude that A + B is a weakly compcact set, and therefore closed. (Sufficiency.) Assume that X is not reflexive Banach space. Then by Theorem of James there exists a continuous linear functional f : X → R, such that ||f || = 1 and f (x) < 1 for every x ∈ B. Let A = {x ∈ X : f (x) 1, ||x|| 2}Then A is a closed bounded and convex set but A + B is not closed.q.e.d.