Optimal isometries for a pair of compact convex subsets of ℝⁿ

Irmina Herburt; Maria Moszyńska

Displaying similar documents to “Optimal isometries for a pair of compact convex subsets of ℝⁿ”

Fractal star bodies

Irmina Herburt, Maria Moszyńska, Dorette Pronk (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 $L_{p}$ metrics for all p ≥ 2 and the symmetric difference metric.

The Demyanov metric and some other metrics in the family of convex sets

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.

A generalization of boundedly compact metric spaces

Gerald Beer, Anna Di Concilio (1991)

Commentationes Mathematicae Universitatis Carolinae

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A metric space $〈 X, d 〉$ is called a $UC$ space provided each continuous function on $X$ into a metric target space is uniformly continuous. We introduce a class of metric spaces that play, relative to the boundedly compact metric spaces, the same role that $UC$ spaces play relative to the compact metric spaces.

Metric enrichment, finite generation, and the path coreflection

Alexandru Chirvasitu (2024)

Archivum Mathematicum

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We prove a number of results involving categories enriched over CMet, the category of complete metric spaces with possibly infinite distances. The category CPMet of path complete metric spaces is locally $ℵ_{1}$ -presentable, closed monoidal, and coreflective in CMet. We also prove that the category CCMet of convex complete metric spaces is not closed monoidal and characterize the isometry- $ℵ_{0}$ -generated objects in CMet, CPMet and CCMet, answering questions by Di Liberti and Rosický. Other results...

On the metric reflection of a pseudometric space in ZF

Horst Herrlich, Kyriakos Keremedis (2015)

Commentationes Mathematicae Universitatis Carolinae

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We show: (i) The countable axiom of choice $𝐂𝐀𝐂$ is equivalent to each one of the statements: (a) a pseudometric space is sequentially compact iff its metric reflection is sequentially compact, (b) a pseudometric space is complete iff its metric reflection is complete. (ii) The countable multiple choice axiom $𝐂𝐌𝐂$ is equivalent to the statement: (a) a pseudometric space is Weierstrass-compact iff its metric reflection is Weierstrass-compact. (iii) The axiom of choice $𝐀𝐂$ is equivalent to each...

The Banach contraction mapping principle and cohomology

Ludvík Janoš (2000)

Commentationes Mathematicae Universitatis Carolinae

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By a dynamical system $(X, T)$ we mean the action of the semigroup $(ℤ^{+}, +)$ on a metrizable topological space $X$ induced by a continuous selfmap $T : X \to X$ . Let $M (X)$ denote the set of all compatible metrics on the space $X$ . Our main objective is to show that a selfmap $T$ of a compact space $X$ is a Banach contraction relative to some $d_{1} \in M (X)$ if and only if there exists some $d_{2} \in M (X)$ which, regarded as a $1$ -cocycle of the system $(X, T) \times (X, T)$ , is a coboundary.

Pairs of convex bodies in a hyperspace over a Minkowski two-dimensional space joined by a unique metric segment

Agnieszka Bogdewicz, Jerzy Grzybowski (2009)

Banach Center Publications

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Let $(ℝ, | | \cdot | |_{)}$ be a Minkowski space with a unit ball and let $ϱ_{H}^{}$ be the Hausdorff metric induced by ${| | \cdot | |}_{}$ in the hyperspace of convex bodies (nonempty, compact, convex subsets of ℝ). R. Schneider [RSP] characterized pairs of elements of which can be joined by unique metric segments with respect to $ϱ_{H}^{B ⁿ}$ for the Euclidean unit ball Bⁿ. We extend Schneider’s theorem to the hyperspace $(², ϱ_{H}^{})$ over any two-dimensional Minkowski space.

Concerning Sets of the First Baire Category with Respect to Different Metrics

Maria Moszyńska, Grzegorz Sójka (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

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We prove that if $ϱ_{H}$ and δ are the Hausdorff metric and the radial metric on the space ⁿ of star bodies in ℝ, with 0 in the kernel and with radial function positive and continuous, then a family ⊂ ⁿ that is meager with respect to $ϱ_{H}$ need not be meager with respect to δ. Further, we show that both the family of fractal star bodies and its complement are dense in ⁿ with respect to δ.

Some properties of convex metric spaces

B. Krakus (1972)

Fundamenta Mathematicae

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On some properties of hyperconvex spaces.

Borkowski, Marcin, Bugajewski, Dariusz, Phulara, Dev (2010)

Fixed Point Theory and Applications [electronic only]

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Characterizing metric spaces whose hyperspaces are homeomorphic to ℓ₂

T. Banakh, R. Voytsitskyy (2008)

Colloquium Mathematicae

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It is shown that the hyperspace $C l d_{H} (X)$ (resp. $B d d_{H} (X)$ ) of non-empty closed (resp. closed and bounded) subsets of a metric space (X,d) is homeomorphic to ℓ₂ if and only if the completion X̅ of X is connected and locally connected, X is topologically complete and nowhere locally compact, and each subset (resp. each bounded subset) of X is totally bounded.

On the symmetric difference metric for convex bodies.

Groemer, H. (2000)

Beiträge zur Algebra und Geometrie

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