Some properties of convex metric spaces
B. Krakus (1972)
Fundamenta Mathematicae
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B. Krakus (1972)
Fundamenta Mathematicae
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J. Bourgain, V.D. Milman (1987)
Inventiones mathematicae
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Lassak, Marek (2008)
Beiträge zur Algebra und Geometrie
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Tadeusz Rzeżuchowski (2012)
Open Mathematics
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We describe some known metrics in the family of convex sets which are stronger than the Hausdorff metric and propose a new one. These stronger metrics preserve in some sense the facial structure of convex sets under small changes of sets.
Inese Bula (2005)
Banach Center Publications
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The paper introduces a notion of strictly convex metric space and strictly convex metric space with round balls. These objects generalize the well known concept of strictly convex Banach space. We prove some fixed point theorems in strictly convex metric spaces with round balls.
Marek Lassak, Monika Nowicka (2010)
Colloquium Mathematicae
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Denote by Kₘ the mirror image of a planar convex body K in a straight line m. It is easy to show that K*ₘ = conv(K ∪ Kₘ) is the smallest by inclusion convex body whose axis of symmetry is m and which contains K. The ratio axs(K) of the area of K to the minimum area of K*ₘ over all straight lines m is a measure of axial symmetry of K. We prove that axs(K) > 1/2√2 for every centrally symmetric convex body and that this estimate cannot be improved in general. We also give a formula for...
Irmina Herburt, Maria Moszyńska (2009)
Banach Center Publications
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In 1989 R. Arnold proved that for every pair (A,B) of compact convex subsets of ℝ there is an Euclidean isometry optimal with respect to L₂ metric and if f₀ is such an isometry, then the Steiner points of f₀(A) and B coincide. In the present paper we solve related problems for metrics topologically equivalent to the Hausdorff metric, in particular for metrics for all p ≥ 2 and the symmetric difference metric.
Dorn, C. (1978)
Portugaliae mathematica
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Taras Banakh, Ivan Hetman (2011)
Studia Mathematica
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We prove that a closed convex subset C of a complete linear metric space X is polyhedral in its closed linear hull if and only if no infinite subset A ⊂ X∖ C can be hidden behind C in the sense that [x,y]∩ C ≠ ∅ for any distinct x,y ∈ A.
A. Zucco (1992)
Discrete & computational geometry
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Lassak, Marek, Nowicka, Monika (2009)
Beiträge zur Algebra und Geometrie
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Tulsi Dass Narang (1981)
Archivum Mathematicum
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