On reduced pairs of bounded closed convex sets.

Jerzy Grzybowski; Ryszard Urbanski

Displaying similar documents to “On reduced pairs of bounded closed convex sets.”

On the number of minimal pairs of compact convex sets that are not translates of one another

J. Grzybowski, R. Urbański (2003)

Studia Mathematica

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Let [A,B] be the family of pairs of compact convex sets equivalent to (A,B). We prove that the cardinality of the set of minimal pairs in [A,B] that are not translates of one another is either 1 or greater than ℵ₀.

$C$ -minimal pairs of compact convex sets.

Pallaschke, D., Urbańska, W., Urbański, R. (1997)

Journal of Convex Analysis

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Pairs of sets with convex union.

Ryszard Urbanski (1997)

Collectanea Mathematica

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In this paper the notion of convex pairs of convex bounded subsets of a Hausdorff topological vector space is introduced. Criteria of convexity pair are proved.

Minimal pairs of bounded closed convex sets

J. Grzybowski, R. Urbański (1997)

Studia Mathematica

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The existence of a minimal element in every equivalence class of pairs of bounded closed convex sets in a reflexive locally convex topological vector space is proved. An example of a non-reflexive Banach space with an equivalence class containing no minimal element is presented.

On the reduction of pairs of bounded closed convex sets

J. Grzybowski, D. Pallaschke, R. Urbański (2008)

Studia Mathematica

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Let X be a Hausdorff topological vector space. For nonempty bounded closed convex sets A,B,C,D ⊂ X we denote by A ∔ B the closure of the algebraic sum A + B, and call the pairs (A,B) and (C,D) equivalent if A ∔ D = B ∔ C. We prove two main theorems on reduction of equivalent pairs. The first theorem implies that, in a finite-dimensional space, a pair of nonempty compact convex sets with a piecewise smooth boundary and parallel tangent spaces at some boundary points is not minimal. The...

On point sets fixing a convex body from within.

Braß, Peter, Herburt, Irmina (1997)

Beiträge zur Algebra und Geometrie

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Minimal pairs of compact convex sets

Diethard Pallaschke, Ryszard Urbański (2004)

Banach Center Publications

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Pairs of compact convex sets naturally arise in quasidifferential calculus as sub- and superdifferentials of a quasidifferentiable function (see Dem86). Since the sub- and superdifferentials are not uniquely determined, minimal representations are of special importance. In this paper we give a survey on some recent results on minimal pairs of closed bounded convex sets in a topological vector space (see PALURB). Particular attention is paid to the problem of characterizing minimal representatives...

General methods of constructing equivalent minimal pairs not unique up to translation.

J. Grzybowski, S. Kaczmarek, R. Urbanski (2000)

Revista Matemática Complutense

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In this paper we give general methods of construction of various equivalent minimal pairs of compact convex sets that are not translates of one another.

Minimal pairs of bounded closed convex sets as minimal representations of elements of the Minkowski-Rådström-Hörmander spaces

Jerzy Grzybowski, Diethard Pallaschke, Ryszard Urbański (2009)

Banach Center Publications

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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.