On point sets fixing a convex body from within.
Braß, Peter, Herburt, Irmina (1997)
Beiträge zur Algebra und Geometrie
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Braß, Peter, Herburt, Irmina (1997)
Beiträge zur Algebra und Geometrie
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Boltyanski, V.G., Morales Amaya, E. (1995)
Journal of Applied Analysis
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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.
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.
Kutateladze, S.S. (2009)
Sibirskij Matematicheskij Zhurnal
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Nicola Fusco, A. Pratelli (2011)
Journal of the European Mathematical Society
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In 1938 Herman Auerbach published a paper where he showed a deep connection between the solutions of the Ulam problem of floating bodies and a class of sets studied by Zindler, which are the planar sets whose bisecting chords all have the same length. In the same paper he conjectured that among Zindler sets the one with minimal area, as well as with maximal perimeter, is the so-called “Auerbach triangle”. We prove this conjecture.
Makeev, V.V. (2005)
Journal of Mathematical Sciences (New York)
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Fradelizi, Matthieu (1999)
Beiträge zur Algebra und Geometrie
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Jerzy Grzybowski, Ryszard Urbanski (2003)
Revista Matemática Complutense
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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.
Carla Peri, Andreana Zucco (1992)
Monatshefte für Mathematik
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Makeev, V.V. (2005)
Journal of Mathematical Sciences (New York)
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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...
Pallaschke, D., Urbańska, W., Urbański, R. (1997)
Journal of Convex Analysis
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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.
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.