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How to get rid of one of the weights in a two-weight Poincaré inequality?

100%
Franchi B., Hajłasz P.
Annales Polonici Mathematici
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2000
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tom 74
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nr 1
97-103
EN
We prove that if a Poincaré inequality with two different weights holds on every ball, then a Poincaré inequality with the same weight on both sides holds as well.
2
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O ilości nierównoważnych metryk

100%
Zieliński M.
Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia
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2017
|
tom 9
139-152
EN
It is well known that there exist many metrics on a non-emptyset. In the case of $(X, \varrho)$ − a finite metric set, it can be easily shown that all the metrics on X are equivalent. This paper examines the number of non-equivalent metrics on uncountably infinite sets.
3
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Compactness in Metric Spaces

100%
Nakasho K., Narita K., Shidama Y.
Formalized Mathematics
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2016
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tom 24
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nr 3
167-172
EN
In this article, we mainly formalize in Mizar [2] the equivalence among a few compactness definitions of metric spaces, norm spaces, and the real line. In the first section, we formalized general topological properties of metric spaces. We discussed openness and closedness of subsets in metric spaces in terms of convergence of element sequences. In the second section, we firstly formalize the definition of sequentially compact, and then discuss the equivalence of compactness, countable compactness, sequential compactness, and totally boundedness with completeness in metric spaces. In the third section, we discuss compactness in norm spaces. We formalize the equivalence of compactness and sequential compactness in norm space. In the fourth section, we formalize topological properties of the real line in terms of convergence of real number sequences. In the last section, we formalize the equivalence of compactness and sequential compactness in the real line. These formalizations are based on [20], [5], [17], [14], and [4].
4
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On Conditions for Unrectifiability of a Metric Space

88%
Hajłasz P., Malekzadeh S.
Analysis and Geometry in Metric Spaces
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2015
|
tom 3
|
nr 1
EN
We find necessary and sufficient conditions for a Lipschitz map f : E ⊂ ℝk → X into a metric space to satisfy ℋk(f(E)) = 0. An interesting feature of our approach is that despite the fact that we are dealing with arbitrary metric spaces, we employ a variant of the classical implicit function theorem. Applications include pure unrectifiability of the Heisenberg groups.
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