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

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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.

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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.

Möbius metric in sector domains

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The Möbius metric $δ_{G}$ is studied in the cases, where its domain $G$ is an open sector of the complex plane. We introduce upper and lower bounds for this metric in terms of the hyperbolic metric and the angle of the sector, and then use these results to find bounds for the distortion of the Möbius metric under quasiregular mappings defined in sector domains. Furthermore, we numerically study the Möbius metric and its connection to the hyperbolic metric in polygon domains.

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.