On inclusion and summands of bounded closed convex sets

Grzybowski, Jerzy; Urbański, Ryszard

doi:10.1007/s10474-005-0020-6

Cited by 3 publications

(1 citation statement)

References 7 publications

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“…Our idea is based on the definition of the Minkowski-Pontryagin difference for convex sets (see [3,4]). Note, that we consider only F-semigroups in this section.…”

Section: The Abstract Differencementioning

confidence: 99%

On Partially Ordered Semigroups and an Abstract Set-Difference

2008

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In this paper we prove the independence of a system of five axioms (S1)-(S5), which was proposed in the book of Pallaschke and Urbański (Pairs of Compact Convex Sets, vol. 548, Kluwer Academic Publishers, Dordrecht, 2002) for partially ordered commutative semigroups. These five axioms (S1)-(S5) are stated in the introduction below. A partially ordered commutative semigroup satisfying these axioms is called a F-semigroup. By the use of a further axiom (S6) we define an abstract difference for the elements of a F-semigroup and prove some basic properties. The most interesting example of a F-semigroup are the nonempty compact convex sets of a topological vector space endowed with the Minkowski sum as operation and the inclusion as partial order. In Section 4 we apply the abstract difference to the problem of minimality of convex fractions.

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“…Our idea is based on the definition of the Minkowski-Pontryagin difference for convex sets (see [3,4]). Note, that we consider only F-semigroups in this section.…”

Section: The Abstract Differencementioning

confidence: 99%

On Partially Ordered Semigroups and an Abstract Set-Difference

2008

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Reduction of finite exhausters

Grzybowski

Pallaschke²,

Urbański

2009

J Glob Optim

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Extreme points of lattice intervals in the Minkowski-Rådström-Hörmander lattice

Grzybowski

Urbański

2008

Proc. Amer. Math. Soc.

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Abstract. In this paper we characterize extreme points of any symmetric interval in the Minkowski-Rådström-Hörmander lattice X over any Hausdorff topological vector space X (Theorem 1). Then we prove that the unit ball in the Minkowski-Rådström-Hörmander lattice X over any normed space X, dimX ≥ 2, has exactly two extreme points (Theorem 2).Let (G, +, ≤) be a commutative lattice-ordered group [1], where x∨y = sup(x, y),In a commutative lattice-ordered group G the following properties hold true:We say that x ∈ G is an extreme point of a subset A of G if for any y, z ∈ A the equality x + x = y + z implies that x = y = z. By ext A we denote the set of all extreme points of A. If G is a vector space, this definition of the extreme point of a subset A coincides with the usual one.The following proposition characterizes extreme points of symmetric intervals in G. a ]. Since x ≤ a, x + ≤ a and x + + x + − a ≤ a; and sinceWe have |z| ≤ a. Since y + z = x + x, x = y = x + + x + − a. Hence x + |x| = x + (x ∨ (−x)) = (x + x) ∨ 0 = (x + x) + = x + + x + = x + a. Then |x| = a.

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On inclusion and summands of bounded closed convex sets

Cited by 3 publications

References 7 publications

On Partially Ordered Semigroups and an Abstract Set-Difference

On Partially Ordered Semigroups and an Abstract Set-Difference

Reduction of finite exhausters

Extreme points of lattice intervals in the Minkowski-Rådström-Hörmander lattice

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