scholarly journals Maximal metric surfaces and the Sobolev-to-Lipschitz property

2020 ◽  
Vol 59 (5) ◽  
Author(s):  
Paul Creutz ◽  
Elefterios Soultanis
Keyword(s):  
Metric Spaces ◽  
Equivalence Classes ◽  
Plateau Problem ◽  
Lipschitz Property ◽  
Maximality Condition ◽  
Volume Rigidity

Abstract We find maximal representatives within equivalence classes of metric spheres. For Ahlfors regular spheres these are uniquely characterized by satisfying the seemingly unrelated notions of Sobolev-to-Lipschitz property, or volume rigidity. We also apply our construction to solutions of the Plateau problem in metric spaces and obtain a variant of the associated intrinsic disc studied by Lytchak–Wenger, which satisfies a related maximality condition.

Cauchy Points of Metric Locales

1989 ◽  
Vol 41 (5) ◽  
pp. 830-854 ◽  
Author(s):  
B. Banaschewski ◽  
A. Pultr
Keyword(s):  
Equivalence Relation ◽  
Metric Spaces ◽  
Equivalence Classes ◽  
Classical Description ◽  
Natural Approach ◽  
Precise Meaning ◽  
Natural Equivalence ◽  
Cauchy Sequences

A natural approach to topology which emphasizes its geometric essence independent of the notion of points is given by the concept of frame (for instance [4], [8]). We consider this a good formalization of the intuitive perception of a space as given by the “places” of non-trivial extent with appropriate geometric relations between them. Viewed from this position, points are artefacts determined by collections of places which may in some sense by considered as collapsing or contracting; the precise meaning of the latter as well as possible notions of equivalence being largely arbitrary, one may indeed have different notions of point on the same “space”. Of course, the well-known notion of a point as a homomorphism into 2 evidently fits into this pattern by the familiar correspondence between these and the completely prime filters. For frames equipped with a diameter as considered in this paper, we introduce a natural alternative, the Cauchy points. These are the obvious counterparts, for metric locales, of equivalence classes of Cauchy sequences familiar from the classical description of completion of metric spaces: indeed they are decreasing sequences for which the diameters tend to zero, identified by a natural equivalence relation.

ON THE COMPLETION OF -METRIC SPACES

2018 ◽  
Vol 98 (2) ◽  
pp. 298-304 ◽  
Author(s):  
NGUYEN VAN DUNG ◽  
VO THI LE HANG
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Quotient Space ◽  
Equivalence Classes ◽  
Cauchy Sequences

Based on the metrisation of $b$-metric spaces of Paluszyński and Stempak [‘On quasi-metric and metric spaces’, Proc. Amer. Math. Soc.137(12) (2009), 4307–4312], we prove that every $b$-metric space has a completion. Our approach resolves the limitation in using the quotient space of equivalence classes of Cauchy sequences to obtain a completion of a $b$-metric space.

scholarly journals Measurability of equivalence classes and MEC$_p$-property in metric spaces

2007 ◽  
pp. 811-830 ◽  
Author(s):  
Esa Järvenpää ◽  
Maarit Järvenpää ◽  
Kevin Rogovin ◽  
Sari Rogovin ◽  
Nageswari Shanmugalingam
Keyword(s):  
Metric Spaces ◽  
Equivalence Classes

scholarly journals Energy and area minimizers in metric spaces

2017 ◽  
Vol 10 (4) ◽  
pp. 407-421 ◽  
Author(s):  
Alexander Lytchak ◽  
Stefan Wenger
Keyword(s):  
Metric Spaces ◽  
Classical Case ◽  
Plateau Problem ◽  
Hölder Exponents ◽  
Area Minimizing ◽  
Definition Of ◽  
Regularity Results ◽  
Quadratic Isoperimetric Inequality ◽  
Energy Minimizers ◽  
Quasi Convexity

AbstractWe show that in the setting of proper metric spaces one obtains a solution of the classical 2-dimensional Plateau problem by minimizing the energy, as in the classical case, once a definition of area has been chosen appropriately. We prove the quasi-convexity of this new definition of area. Under the assumption of a quadratic isoperimetric inequality we establish regularity results for energy minimizers and improve Hölder exponents of some area-minimizing discs.

scholarly journals Coarse compactifications and controlled products

2020 ◽  
pp. 1-26
Author(s):  
Tomohiro Fukaya ◽  
Shin-ichi Oguni ◽  
Takamitsu Yamauchi
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Equivalence Classes ◽  
Ideal Boundary ◽  
Bijective Correspondence ◽  
Gromov Boundary ◽  
The Ideal

We introduce the notion of controlled products on metric spaces as a generalization of Gromov products, and construct boundaries by using controlled products, which we call the Gromov boundaries. It is shown that the Gromov boundary with respect to a controlled product on a proper metric space is the ideal boundary of a coarse compactification of the space. It is also shown that there is a bijective correspondence between the set of all coarse equivalence classes of controlled products and the set of all equivalence classes of coarse compactifications.

THE TUKEY ORDER ON COMPACT SUBSETS OF SEPARABLE METRIC SPACES

2016 ◽  
Vol 81 (1) ◽  
pp. 181-200 ◽  
Author(s):  
PAUL GARTSIDE ◽  
ANA MAMATELASHVILI
Keyword(s):  
Partially Ordered Set ◽  
Metric Spaces ◽  
Ordered Set ◽  
Equivalence Classes ◽  
Order Structure ◽  
Partially Ordered ◽  
Image Position ◽  
Separable Metrizable Space ◽  
Directed Sets ◽  
Compact Subsets

AbstractOne partially ordered set, Q, is a Tukey quotient of another, P, if there is a map (a Tukey quotient) $\phi :P \to Q$ carrying cofinal sets of P to cofinal sets of Q. Two partial orders which are mutual Tukey quotients of each other are said to be Tukey equivalent. Let ${\cal D}_{\rm{}} $ be the partially ordered set of Tukey equivalence classes of directed sets of size $ \le {\rm{}}$. It is shown that ${\cal D}_{\rm{}} $ contains an antichain of size $2^{\rm{}} $, and so has size $2^{\rm{}} $. The elements of the antichain are of the form ${\cal K}\left( M \right)$, the set of compact subsets of a separable metrizable space M, ordered by inclusion. The order structure of such ${\cal K}\left( M \right)$’s under Tukey quotients is investigated. Relative Tukey quotients are introduced. Applications are given to function spaces and to the complexity of weakly countably determined Banach spaces and Gul’ko compacta.

Crystallography in two-dimensional metric spaces*

1969 ◽  
Vol 130 (1-6) ◽  
pp. 277-303 ◽  
Author(s):  
Aloysio Janner ◽  
Edgar Ascher
Keyword(s):  
Metric Spaces ◽  
Two Dimensional

Solution of Volterra integral inclusion in b-metric spaces via new fixed point theorem

2016 ◽  
Vol 2017 (1) ◽  
pp. 17-30 ◽  
Author(s):  
Muhammad Usman Ali ◽  
◽  
Tayyab Kamran ◽  
Mihai Postolache ◽  
◽  
...  
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Metric Spaces ◽  
Integral Inclusion

ON THE QUANTIFICATION OF UNIFORM PROPERTIES.PART 2

2001 ◽  
Vol 37 (1-2) ◽  
pp. 169-184
Author(s):  
B. Windels
Keyword(s):  
Metric Spaces ◽  
Uniform Spaces ◽  
Complete Metric Spaces ◽  
The Subject

In 1930 Kuratowski introduced the measure of non-compactness for complete metric spaces in order to measure the discrepancy a set may have from being compact.Since then several variants and generalizations concerning quanti .cation of topological and uniform properties have been studied.The introduction of approach uniform spaces,establishes a unifying setting which allows for a canonical quanti .cation of uniform concepts,such as completeness,which is the subject of this article.

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