Questions tagged [metric-spaces]
A metric space is a pair $(X,d)$, where $X$ is a set and $d:X \times X \to \mathbb{R}$ satisfies the following conditions for all $x,y,z \in X$. (Symmetry) $d(x,y)=d(y,x)$. (Identity of Indiscernibles) $d(x,y)=0$ if and only if $x=y$. (Triangle Inequality) $d(x,y)+d(y,z) \geq d(x,z)$.
442 questions
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$\omega$-homogenous space which is not $\omega_1$-homogenous
Consider a metric space $(X,d)$ and let $\kappa$ be a cardinal. We say that $(X,d)$ is $\kappa$-homogenous, if every (surjective) isometry $h:X_1 \to X_2$ between subspaces of $(X,d)$ of size $< \...
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Get an estimate on $L^{2}(0,1)$ [closed]
Consider $f \in L^{2}(0,1)$ and $g \in L^{\infty}(0,1)$ such that
$ \text{lim} ~g(x) = 0 \ \ \text{when} \ \ x \to 0^{+};$
$g(x) > 0 \ \forall x \in (0,1)$;
$\text{lim}~\dfrac{g(x)}{x^{\alpha}} =...
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separable support of Borel measure, with tau-additive measure and full support
I have a problem with Proposition 7.2.10 in Bogachev's Measure Theory Volume II book on page 77 (I have link to my drive with that book https://drive.google.com/file/d/...
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Dense $G_{\delta}$ set with $\sigma$-porous complement is cofinite?
Let $X$ be a separable Banach space and $D\subseteq X$ be a
proper, connected, and dense $G_{\delta}$ subset of $X$,
$X-D$ is $\sigma$-porous.
Then is $X-D$ contained in a finite-dimensional ...
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If a function follows another one's range order, can we say it follows some continuity properties?
Suppose that $(X,d)$ be a complete metric space, $f:X \to \mathbb{R}$ is a sequentially lower monotone function. Let $g: X \to \mathbb{R}$ be a function with the property: $f(x)\leq f(y) \Rightarrow g(...
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Graphs represented by a subset of a metric space
Let $(X,d)$ be a metric space, and suppose $S\subseteq X$ is a finite subset in which all pairwise distances are distinct (formal definition here).
If $x\in S$ and $k$ is a non-negative integer with $...
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825
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Approximation of semicontinuous function
I'm looking for a reference for the theorem saying that a real-valued lower (upper) semicontinuous function on any metric space can be reached as a pointwise limit by a non-decreasing (non-increasing) ...
user114232
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Covering numbers of uniformly bounded subsets of Gromov-Hausdorff space
For any metric space $X$ and $\varepsilon>0$, let $$\text{cov}(X,\varepsilon)=\min\{n\,|\,X\text{ has a cover by }n\text{ many closed }\varepsilon\text{-balls}\},$$
be the ordinary covering ...
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Is there any result concerning on the metric dimension of inverse limit?
To be specific, my question is as follows:
Question: Let $X$ be an inverse limit of compact metric spaces $(X_i, d_i)$, then does it hold
$\dim(X, d) \leq \sup_i \{\dim (X_i, d_i)\}$ for some ...
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Does smallness of Gromov-Hausdorff distance on scale 2 imply smallness on GH distance on scale 1?
Let $(M,g)$ be a Riemannian manifold and $C(Y)$ be a metric cone over $Y$. Let $B_r$ denote the geodesic ball of radius $r$ centered at a fixed point $x$ in $M$ and $C_r$ denote the metric ball of ...
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Random covering on rectangles
Let $\mathrm{Rect}$ denote the class of axis-parallel rectangles $r: \mathbb{R}^2 \to \{0,1\}$, assigning $1$ if the point is inside the rectangle and $0$ otherwise. Let $\mathcal{D}$ be a ...
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Definition of semi-metric for empirical process theory
In the following lecture notes on empirical processes (https://www.stat.columbia.edu/~bodhi/Talks/Emp-Proc-Lecture-Notes.pdf) a semi-metric space $(\Theta, d)$ is defined in the following way: for any ...
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Any useful bases for the topology induced by the $t$-Wasserstein distance?
I am working on $\mathbb R ^d$ equipped with the usual Euclidean metric. I know of one nice base for $\mathcal W _t$, namely: $$\left\{ B_p (r) : r>0, p=\sum_{i=1} ^n \alpha_i \delta_{x_i},\text{ ...
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Wasserstein space isomorphic to original space?
Is there a complete measurable metric space $(X,d)$ for which its $p$-Wasserstein space $W(X)$ is isometrically isomorphic to $(X,d)$ for some $p \in [1,\infty]$?
Note that there is a canonical non-...
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The set of continuous bounded functions $f:X\to Y$ is dense in $L^p(X,Y)$ where $X,Y$ are Polish
It is well known that the set of real-valued continuous functions with compact support is dense in $L^p(\mu)$ where $\mu$ is a Radon measure (see e.g. [Folland, Proposition 7.9]) Clearly, the set of ...
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Example of a metrizable space that is not an ANR
I have been looking for an example of a metrizable space that is not an absolute neighborhood retract (ANR).
Recall that a metrizable space $X$ is called an ANR if there exists an open set $U$ in a ...
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What is a metric for weak convergence of finite measures on a non compact, complete and separable metric space?
Consider the set of finite positive measures on a complete, separable, but not compact, metric space $S$, endowed with the topology under which a sequence of finite positive measures $\{\mu_n\}$ ...
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Properties of doubling metric spaces
At present I work with tools that involves doubling metric space, my definition of DME is:
A metric space $X$ is called doubling with constant $N$, where $N \geq 1$ is an integer, if, for each ball $...
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131
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Cyclic group action and finite invariant set
Let $(X, d)$ be a compact metric space and $G$ a discrete group acting on $X$ such that, for each $g\in G$, the mapping $x\mapsto g\cdot x$ defines a homeomorphism on $X$
Is it true that the ...
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A generalization about the density of $\mathcal C_c(X, E)$ in $\mathcal L_p (X, \mu, E)$ when $E$ is a Banach space
Let $X$ be a metric space, $\mu$ a $\sigma$-finite non-negative Borel measure on $X$, and $(E, |\cdot|)$ a Banach space. Let $\mathcal L_p := \mathcal L_p (X, \mu, E)$ and $\|\cdot\|_{\mathcal L_p}$ ...
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Complex Borel measures: relation between the total variation norm of a measure and those of its real and imaginary parts
Let $X$ be a metric space and $\mathcal B$ its Borel $\sigma$-algebra. For $B \in \mathcal B$ we denote by $\Pi(B)$ the collection of all finite measurable partitions of $B$, i.e.,
$$
\Pi(B)=\left\{\...
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Let $E$ be Banach, $\mu_n\to\mu$ weakly on a locally compact $X$, and $f \in C_b(X, E)$. Does $\int f\mathrm d\mu_n\to\int f\mathrm d\mu$ in norm?
Let
$X$ be a metric space,
$(E, |\cdot|)$ a Banach space
$\mathcal M(X)$ the space of all finite signed Borel measures on $X$,
$\mathcal C_b(X)$ be the space of real-valued bounded continuous ...
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49
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When can a compact metric space be covered by finitely many nearly-disjoint closed and convex sets?
This question is a follow-up of the following negative question.
Let $(X,d)$ be a (non-empty) compact metric space.
More generally than in the first post, I'll call a set of non-empty subsets $C_1,\...
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Terminology: maps which are bi-Lipschitz on compact subsets
Let $X$ and $Y$ be metric spaces and let $f:X\rightarrow Y$ be such that: for every compact subset $K$ of $X$ the restricted map $f|_K:K\rightarrow Y$ defined by $f|_K(x)=f(x)$ is bi-Lipschitz (with ...
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Finite sets are residual in the Hausdorff space
Let $X$ be a metric space, let $\mathbb{H}(X)$ denote the set of non-empty closed subsets of $X$ with Hausdorff metric which we denote by $d_{\mathbb{H}(X)}$, and let $\mathbb{H}_{\operatorname{fin}}(...
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Do all manifolds admit metrics with Euclidean balls?
Let $M$ be a compact topological n-manifold. Suppose we are given a locally flat embedding $M \subset \mathbb{R}^{n+k}$. This induces a metric on $M$ by restriction. Is it true that for $\epsilon$ ...
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Terminology "upper" Ahlfors regular measure
Let $(X,d)$ be a metric space and $m$ be a Borel measure on $(X,d)$. The measure $m$ is called Ahlors regular if $m(B(x,r))\asymp r^q$ for some $q>0$ and each $x\in X$. Is there a name for ...
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Holder-continuous barycenter maps
Let $(X,d)$ be a complete locally-compact metric space. We define the $p$-barycenter map as a continuous function:
$$
\beta:\mathcal{P}_p(X)\rightarrow X,
$$
which is a right-inverse of the map ...
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Banach fixed point theorem / convergence squeeze
I am trying to prove a convergence result on an iterative scheme which has the initial point defined as
$$x_1 = \frac{1 - s(x_0)}{s(x_0)}$$
where s(x) is some unknown function.
Here is my theorem and ...
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Gromov–Hausdorff closure of non-positively curved graphs
Setup:
Let $\Gamma$ be the set of non-positively curved weighted connected graphs, with finitely many points, which are isometrically embedded in $\mathbb{R}^n$; for some $n\in \mathbb{N}$;$n\geq 2$. ...
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Does the lemma remain valid in b-metric space?
Let $(X,d)$ be a complete metric space.
$$CB(X)=\{A : A \text{ is a nonempty closed and bounded subset of }X \},$$
$$D(A,B)=\inf \{d(a,b) : a\in A , b\in B\},$$
$$\sigma (A,B)=\sup \{d(a,b) : a\in A , ...
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A closed subset $B$ of the Hilbert cube such that $\operatorname{Int}(B) = \emptyset$ and $B$ is not a z-set
$\def\cube{\mathbf Q}$Let $\cube$ be the Hilbert cube. Give an example of a closed subset $B$ of $\cube$ such that $\operatorname{Int}(B) = \emptyset$ and $B$ is not a z-set.
If anyone has any idea ...
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Skorohod Space with $J_1$ topology homeomorphic to Frechet Space
Is the Skorohod space $D([0,T];\mathbb{R}^d)$ equipped with the $J_1$ topology homoeomorphic to a separable Fr\'{e}chet space. In particular, is it homeomorphic to $L_{\mu}^1(\mathcal{B}([0,1])$ ...
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Contractable and Simply Connected Doubling Spaces Homeomorphic to Euclidean Space
Is there a characterization of all simply connected, contractable doubling metric spaces which are homeomorphic to a simply connected subset of Euclidean space?
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An extension for lower semi continuous lower bounded real valued functions class
Let $(X,d)$ be a complete metric space. I need some explanations about the class of all functions like $f$ which have $f:X \to \mathbb{R}\cup\{ +\infty\}$, be a lower bounded and, for all $y \in X$ we ...
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Lipschitz extensions preserving the convex hull of the range
We assume that $X$ is a metric space and that $A \subseteq X$ is a subset. Let $f : A \rightarrow \mathbb R$ be a Lipschitz-continuous function with Lipschitz constant $L$.
By the Kirszbraun theorem, ...
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68
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Estimate bounds on Minkowski distance from point to a segment in Lp space
Assumptions
Let
$L_p(x,y)=(\sum_i|x_i - y_i|^p)^{1/p}$ (Minkowski metric),
$a,b$ be arbitrary $n$-dimensional points,
$c$ be a point that satisfies $L_p(a,b) = L_p(a,c) + L_p(c,b)$, i.e., a point ...
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3
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502
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Metric-space with a ball inside a smaller ball [closed]
Could you tell me an example to an $(X,\varrho)$ metric-space with balls $B(x_1,r_1)$ and $B(x_2,r_2)$ where $r_1<r_2$ but also $B(x_2,r_2)\subset B(x_1,r_1)$?
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In a metrizable order topology, is the order relation compatible with the metric? [closed]
Does $x \le y \le z$ imply $d(x, y) \le d(x, z)$?!
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1
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99
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Existence of continuous selection for metric projection
Let $(X,d)$ be a separable complete geodesic metric space and let $K$ be a compact (non-empty) subset of $X$. Without assuming things like linearity, the convexity of $K$, and locally convexity, ...
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1
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141
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Interpretation and validity of modified Heisenberg uncertainty principle in a metric context? [closed]
Considering the Heisenberg uncertainty principle, which states $\Delta x \cdot \Delta p \geq h$, I've explored a modified version by computing $(\Delta x + 1)(\Delta p + 1) \geq \Delta x \cdot \Delta ...
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Can a Polish space have two different topologies?
Let $X$ be a Polish space with the compatible metric being $d_1$. So $(X,d_1)$ is a separable complete metric space, and the topology is generated by $d_1$.
Can there be a metric $d_2$ such that $(X,...
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