Median for metric spaces

Nacereddine Belili; Henri Heinich

Displaying similar documents to “Median for metric spaces”

On the tangency of sets in generalized metric spaces for certain functions of the class $F_{ρ}^{*}$

T. Konik (1991)

Matematički Vesnik

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Wijsman hyperspaces of non-separable metric spaces

Rodrigo Hernández-Gutiérrez, Paul J. Szeptycki (2015)

Fundamenta Mathematicae

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Given a metric space ⟨X,ρ⟩, consider its hyperspace of closed sets CL(X) with the Wijsman topology $τ_{W (ρ)}$ . It is known that $⟨ C L (X), τ_{W (ρ)} ⟩$ is metrizable if and only if X is separable, and it is an open question by Di Maio and Meccariello whether this is equivalent to $⟨ C L (X), τ_{W (ρ)} ⟩$ being normal. We prove that if the weight of X is a regular uncountable cardinal and X is locally separable, then $⟨ C L (X), τ_{W (ρ)} ⟩$ is not normal. We also solve some questions by Cao, Junnila and Moors regarding isolated points in Wijsman hyperspaces. ...

Isometric embeddings of a class of separable metric spaces into Banach spaces

Sophocles K. Mercourakis, Vassiliadis G. Vassiliadis (2018)

Commentationes Mathematicae Universitatis Carolinae

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Let $(M, d)$ be a bounded countable metric space and $c > 0$ a constant, such that $d (x, y) + d (y, z) - d (x, z) \geq c$ , for any pairwise distinct points $x, y, z$ of $M$ . For such metric spaces we prove that they can be isometrically embedded into any Banach space containing an isomorphic copy of $ℓ_{\infty}$ .

The nonexistence of universal metric flows

Stefan Geschke (2018)

Commentationes Mathematicae Universitatis Carolinae

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We consider dynamical systems of the form $(X, f)$ where $X$ is a compact metric space and $f : X \to X$ is either a continuous map or a homeomorphism and provide a new proof that there is no universal metric dynamical system of this kind. The same is true for metric minimal dynamical systems and for metric abstract $ω$ -limit sets, answering a question by Will Brian.

Some approximate fixed point theorems without continuity of the operator using auxiliary functions

Sumit Chandok, Arslan Hojjat Ansari, Tulsi Dass Narang (2019)

Mathematica Bohemica

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We introduce partial generalized convex contractions of order $4$ and rank $4$ using some auxiliary functions. We present some results on approximate fixed points and fixed points for such class of mappings having no continuity condition in $α$ -complete metric spaces and $μ$ -complete metric spaces. Also, as an application, some fixed point results in a metric space endowed with a binary relation and some approximate fixed point results in a metric space endowed with a graph have been obtained....

The discriminant and oscillation lengths for contact and Legendrian isotopies

Vincent Colin, Sheila Sandon (2015)

Journal of the European Mathematical Society

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We define an integer-valued non-degenerate bi-invariant metric (the discriminant metric) on the universal cover of the identity component of the contactomorphism group of any contact manifold. This metric has a very simple geometric definition, based on the notion of discriminant points of contactomorphisms. Using generating functions we prove that the discriminant metric is unbounded for the standard contact structures on $ℝ^{2 n} \times S^{1}$ and $ℝ P^{2 n + 1}$ . On the other hand we also show by elementary arguments...

Adjacent dyadic systems and the $L^{p}$ -boundedness of shift operators in metric spaces revisited

Olli Tapiola (2016)

Colloquium Mathematicae

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With the help of recent adjacent dyadic constructions by Hytönen and the author, we give an alternative proof of results of Lechner, Müller and Passenbrunner about the $L^{p}$ -boundedness of shift operators acting on functions $f \in L^{p} (X; E)$ where 1 < p < ∞, X is a metric space and E is a UMD space.

Metric unconditionality and Fourier analysis

Stefan Neuwirth (1998)

Studia Mathematica

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We investigate several aspects of almost 1-unconditionality. We characterize the metric unconditional approximation property (umap) in terms of “block unconditionality”. Then we focus on translation invariant subspaces $L_{E}^{p} ()$ and $C_{E} ()$ of functions on the circle and express block unconditionality as arithmetical conditions on E. Our work shows that the spaces $p_{E} ()$ , p an even integer, have a singular behaviour from the almost isometric point of view: property (umap) does not interpolate between $L_{E}^{p} ()$ ...

Generalized Lebesgue points for Sobolev functions

Nijjwal Karak (2017)

Czechoslovak Mathematical Journal

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In many recent articles, medians have been used as a replacement of integral averages when the function fails to be locally integrable. A point $x$ in a metric measure space $(X, d, μ)$ is called a generalized Lebesgue point of a measurable function $f$ if the medians of $f$ over the balls $B (x, r)$ converge to $f (x)$ when $r$ converges to $0$ . We know that almost every point of a measurable, almost everywhere finite function is a generalized Lebesgue point and the same is true for every point of a continuous function....

Asymptotic rate of convergence in the degenerate U-statistics of second order

Olga Yanushkevichiene (2010)

Banach Center Publications

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Let X,X₁,...,Xₙ be independent identically distributed random variables taking values in a measurable space (Θ,ℜ ). Let h(x,y) and g(x) be real valued measurable functions of the arguments x,y ∈ Θ and let h(x,y) be symmetric. We consider U-statistics of the type $T (X ₁, . . ., X ₙ) = n^{- 1} \sum_{1 \leq i L e t q_{i} (i \geq 1) b e e i g e n v a l u e s o f t h e H i l b e r t - S c h m i d t o p e r a t o r a s s o c i a t e d w i t h t h e k e r n e l h (x, y), a n d q ₁ b e t h e l a r g e s t i n a b s o l u t e v a l u e o n e . W e p r o v e t h a t}$ Δn = ρ(T(X₁,...,Xₙ),T(G₁,..., Gₙ)) ≤ (cβ’1/6)/(√(|q₁|) n1/12) $,$ where $G_{i}$ , 1 ≤ i ≤ n, are i.i.d. Gaussian random vectors, ρ is the Kolmogorov (or uniform) distance and $β^{'} : = E | h (X, X ₁) | ³ + {E | h (X, X ₁) |}^{18 / 5} + E | g (X) | ³ + {E | g (X) |}^{18 / 5} + 1 < \infty$ .

Wasserstein metric and subordination

Philippe Clément, Wolfgang Desch (2008)

Studia Mathematica

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Let $(X, d_{X})$ , $(Ω, d_{Ω})$ be complete separable metric spaces. Denote by (X) the space of probability measures on X, by $W_{p}$ the p-Wasserstein metric with some p ∈ [1,∞), and by $_{p} (X)$ the space of probability measures on X with finite Wasserstein distance from any point measure. Let $f : Ω \to_{p} (X)$ , $ω \mapsto f_{ω}$ , be a Borel map such that f is a contraction from $(Ω, d_{Ω})$ into $(_{p} (X), W_{p})$ . Let ν₁,ν₂ be probability measures on Ω with $W_{p} (ν ₁, ν ₂)$ finite. On X we consider the subordinated measures $μ_{i} = \int_{Ω} f_{ω} d ν_{i} (ω)$ . Then $W_{p} (μ ₁, μ ₂) \leq W_{p} (ν ₁, ν ₂)$ . As an application we show that the solution measures $ϱ_{α} (t)$ ...

Differential equations in metric spaces

Jacek Tabor (2002)

Mathematica Bohemica

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We give a meaning to derivative of a function $u ℝ \to X$ , where $X$ is a complete metric space. This enables us to investigate differential equations in a metric space. One can prove in particular Gronwall’s Lemma, Peano and Picard Existence Theorems, Lyapunov Theorem or Nagumo Theorem in metric spaces. The main idea is to define the tangent space $𝒯_{x} X$ of $x \in X$ . Let $u, v [0, 1) \to X$ , $u (0) = v (0)$ be continuous at zero. Then by the definition $u$ and $v$ are in the same equivalence class if they are tangent at zero, that is if $lim_{h \to 0^{+}} \frac{d (u (h), v (h))}{h} = 0 .$ By...

Some limit theorems for $m$ -pairwise negative quadrant dependent random variables

Yongfeng Wu, Jiangyan Peng (2018)

Kybernetika

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The authors first establish the Marcinkiewicz-Zygmund inequalities with exponent $p$ ( $1 \leq p \leq 2$ ) for $m$ -pairwise negatively quadrant dependent ( $m$ -PNQD) random variables. By means of the inequalities, the authors obtain some limit theorems for arrays of rowwise $m$ -PNQD random variables, which extend and improve the corresponding results in [Y. Meng and Z. Lin (2009)] and [H. S. Sung (2013)]. It is worthy to point out that the open problem of [H. S. Sung, S. Lisawadi, and A. Volodin (2008)] can be...

Multi-variate correlation and mixtures of product measures

Tim Austin (2020)

Kybernetika

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Total correlation (‘TC’) and dual total correlation (‘DTC’) are two classical ways to quantify the correlation among an $n$ -tuple of random variables. They both reduce to mutual information when $n = 2$ . The first part of this paper sets up the theory of TC and DTC for general random variables, not necessarily finite-valued. This generality has not been exposed in the literature before. The second part considers the structural implications when a joint distribution $μ$ has small TC or DTC. If...