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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Does the topology of Wasserstein space $(\mathcal P_p (E), W_p)$ coincide with the initial topology induced by $\mathcal C_b(E) \cup \{g_p\}$?
Let $(E, d)$ be a Polish space and $\mathcal C_b(E)$ the space of all real-valued bounded continuous functions on $E$. Let $p \in [1, \infty)$ and $\mathcal P_p (E)$ the space of all Borel probability ...
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Conditions that ensure the metric topology of $E$ coincides with the initial topology induced by a collection of real-valued functions on $E$
Let $(E, d)$ be a metric space and $\mathcal F$ a collection of real-valued functions on $E$. Assume that for all $x,x_n \in E$ with $n\in \mathbb N$,
$$
x_n \to x \iff [f(x_n) \to f(x) \quad \forall ...
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2
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Optimal transport: the existence of an optimal pair of $c$-conjugate functions
$\newcommand{\diff}{ \, \mathrm d}$
Let
$X,Y$ be Polish spaces,
$\mathcal C_b(X)$ the space of all real-valued bounded continuous functions on $X$,
$\mathcal P(X)$ the space of Borel probability ...
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A variant of disintegration theorem where the assumptions on $f$ and $g$ are exchanged
I have recently read about about disintegration theorem, i.e.,
Disintegration theorem Let
$X$ be a Polish space, $\mathcal X$ its Borel $\sigma$-algebra, and $\mu$ a Borel probability measure on $X$...
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158
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Does "achieving more GH-distances than some compact space" imply compactness?
Previously asked and bountied at MSE:
For complete metric spaces $X,Y$, write $X\trianglelefteq Y$ iff for every complete metric space $Z$ such that the Gromov-Hausdorff distance between $X$ and $Z$ ...
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2
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Reference request: "Tangent relation" in metric spaces
Let $X,Y$ be metric spaces. Let $f,g : X \to Y$ be two maps and $x_0 \in X$. Let us say that $f$ and $g$ are tangent at $x_0$ if the following condition is satisfied: For every $\epsilon > 0$ there ...
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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: does $\mu_n \to \mu$ weakly imply $|\mu|(\Theta) \le \liminf_n |\mu_n|(\Theta)$ for every open subset $\Theta$?
Let
$\Omega$ be a metric space,
$C_b(\Omega)$ the space of all real-valued bounded continuous functions on $\Omega$, and
$\mathcal{M}(\Omega)$ the space of all finite signed Borel measures on $\Omega$...
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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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Vague convergence: confusion about the regularity of a signed Radon measure and that of its variation
I'm reading a proof of below theorem from this paper.
Theorem A.3. Let $\Omega$ be a locally compact normal Hausdorff space. Let $\left\{\mu_n\right\} \cup\{\mu\} \subset \mathcal{M}(\Omega)$ and ...
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Results in computational geometry utilizing doubling dimension of a metric space
According to Wikipedia,
However, many results from classical harmonic analysis and computational geometry extend to the setting of metric spaces with doubling measures.
My question is: what are some ...
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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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Portmanteau theorem for finite signed Borel measures
Let
$X$ be a metric space,
$\mathcal M(X)$ the space of all finite signed Borel measures on $X$,
$\mathcal M_+(X)$ the space of all finite nonnegative Borel measures on $X$,
$\mathcal M_1(X)$ the ...
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1
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1
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Are there some conditions on a metric space $X$ such that these two types of weak converge of finite signed Borel measures on $X$ are related?
Let
$X$ be a metric space,
$\mathcal M(X)$ the space of all finite signed Borel measures on $X$, and
$\mathcal C_b(X)$ be the space of real-valued bounded continuous functions on $X$.
Then $\mathcal ...
- 835
6
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1
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257
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Expected doubling constant of a random Erdős–Rényi graph
Consider the $G(n,p)$ random graph model where $n$ is a ``large'' positive integer and $p\in (0,1)$. We may equip every realized random graph $G$ with its shortest path distance, making it into a (...
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Extending a partially defined metric on a metrizable space
Let $X$ be a metrizable topological space, $A\subseteq X\times X$ a nonempty closed subset which is reflexive, symmetric and transitive, $d:A\to \mathbb{R}_+$ a continuous function that satisfies the ...
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681
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Uniform density of Lipschitz maps is space of continuous function — for general metric spaces
Let $X$ and $Y$ be metric space, $X$ be compact, $C(X,Y)$ denote the set of continuous functions from $X$ to $Y$ with uniform convergence on compacts topology, and $\operatorname{Lip}(X,Y)$ denote the ...
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Disintegration of measures: a confusion about an existence proof from a lecture note
I'm reading a proof of Theorem 2.25 below from this note. First, we recall a definition and a theorem, i.e.,
Theorem 2.25 (Disintegration). Let $\left(Z, d_Z\right)$ and $\left(X, d_X\right)$ be ...
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How to find the point at minimal average distance of a given measure
Given a compactly supported probability measure $m$ on $\mathbb{R}^n$, we can define its average distance to a point $x$ as $\int_\mathbb{R^n}d(x,y)dm(y)$. In this question I found that for a given ...
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m-point-homogeneous, but not (m+1)-point-homogeneous
It is straightforward to check that the discrete cube $Q=\{0,1\}^n$ with $\ell^1$-metric is 3-point-homogeneous, but not 4-point-homogeneous (assuming $n$ is large).
In other words, if $A\subset Q$ ...
- 45k
4
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1
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159
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Extending a metric in a bi-Lipschitz way
Suppose we are in the following situation: $(X,d)$ is a metric space and $Y$ is a subspace of $X$. Furthermore we have a different metric $\delta$ defined on $Y$ such that $\delta$ is bi Lipschitz ...
0
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1
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61
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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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Partitioning a smooth manifold into geodesically convex sets
Let $X$ be a connected and compact $d$-dimensional smooth manifold; where $d$ is a positive integer. Does (or rather, when does) there exist a metric $\rho$ on $X$ generating $X$'s topology and a ...
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"Geodesic coherent" partition of a graph
Let $G=(V,E)$ be a finite undirected graph which we equip with its usual graph geodesic distance $d_G$ making $(G,d_G)$ into a metric space; let $1<\#V<\infty$. For a given $1<N< \#V$ ...
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Definition of the category QMet of metric spaces and quasi-isometries
I am following Clara Löh's Geometric Group Theory. An Introduction, and in remark 5.1.12, she defines the category QMet whose objects are metric spaces and whose morphisms are quasi-isometric ...
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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^{+};$
$g(x) > 0 \ \forall x \in (0,1)$;
$\text{lim}~\dfrac{g(x)}{x^{\alpha}} =...
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2
answers
231
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$(1+\epsilon)$-bilipschitz parametrization of Lipschitz manifold
Let $\mathscr{H}^m$ be the $m$ dimensional Hausdorff measure in $\mathbb{R}^n$, $m\leq n$. Is it true that for $\mathscr{H}^m$-almost every point $p$ on a Lipschitz manifold $M$ of dimension $m$ ...
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158
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Constant in Naor and Neiman's Assouad Theorem
In Naor and Neiman's Assouad embedding theorem - "Assouad’s theorem with dimension independent of the snowflaking" Revisita Mathematica, the authors derive quantitative estimates on the ...
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424
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Lipschitz mappings, covering dimension
Is there a compact metric space $X$ of covering dimension $2$ without a Lipschitz surjection on $[0,1]^2$?
For a space $X$ with Hausdorff dimension greater than $2$, we have a negative answer (see ...
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115
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Isometric embeddings of $c_0$ into metric spaces
Are there any nice and useful criteria or theorems which assert when a given metric space $M$ contains an isometric (not necessarily linear) copy of the Banach space $c_0$ or its unit ball $B_{c_0}$? (...
- 1,151
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1
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514
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Distance between two points using triangulation
Suppose we have two points $p_1$ and $p_2$ in a metric space with unknown dimensionality, with no way to directly compute the distance between them, e.g. no coordinates.
Say we can randomly sample a ...
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562
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What do category theorists know about "probabilistic metric spaces"?
I recently stumbled upon the notion of probabilistic metric space as a generalization of Lawvere's metric spaces, and I am very interested in understanding it deeper.
In short, instead of a distance $...
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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,\...
- 5,405
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Are normal metric currents dense in the space of all metric currents?
It is classical that Euclidean normal currents are dense in the space of all currents.
This can be achieved through mollification.
What I want to know if this is still true for metric currents.
In ...
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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 ...
- 5,405
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0
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97
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Are Hölder functions between Banach spaces residual in the compact-open topology?
Let $X$ and $Y$ be Banach spaces and let $C(X,Y)$ be the set of continuous functions from $X$ to $Y$ equipped with the topology of uniform convergence on compact sets (i.e. the compact-open topology). ...
- 5,405
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0
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113
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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}}(...
- 5,405
4
votes
1
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407
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Lipschitz-regularity of partition of unity
Let $K$ be a compact subset of $\mathbb{R}^n$ and $\mathcal{U}$ be a finite collection of open subsets covering $K$ satisfying the minimality property: for every $U\in \mathcal{U}$, the sub-collection ...
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3
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2
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226
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Linear process close to a Gaussian process
A linear process $(X_t)_{t \in \mathbb{Z}}$ is usually written as a moving-average process with infinity order:
\begin{equation}\label{linear_process}\tag{Eq. 1.1}
X_{t} = \sum_{j =0 }^\infty \...
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0
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238
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Example of CAT($k$) space [closed]
Good time of day. I repeat the question from MSE (https://math.stackexchange.com/questions/4464888/question-about-example-of-catk-space) because no response has been received.Question is the following:...
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0
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449
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Bound on covering number of Lipschitz functions – missing part in proofs of Kolmogorov et al
Given a metric space $(\mathcal{X},\rho)$ and $\mathcal{A}\subset\mathcal{X}$ totally bounded, i.e. $\mathcal{A}$ has a finite $\varepsilon$-covering for any $\varepsilon>0$. Consider $\...
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279
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Product topology from two premetric spaces induced by sum of premetrics?
For metric spaces $(M_1, d_1)$ and $(M_2, d_2)$, it is an exercise that the product topology on $M_1\times M_2$ is induced by the metric $d((x_1, y_1), (x_2, y_2)) =d_1(x_1, x_2) + d_2(y_1, y_2)$.
Do ...
- 463
3
votes
1
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486
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There exists differentiable curves arbitrarily close to the continuous ones
Let $M$ be a Riemannian manifold; if $d$ is the distance on $M$, we can consider the distance $D$ between any two continuous curves given by $D(c, \gamma) = \max _{t \in [0,1]} d(c(t), \gamma(t))$.
...
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10
votes
1
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561
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Does a compact contractible metric space have a point that is fixed by all isometries?
Let $(X,d)$ be a compact and contractible metric space. Let $\operatorname{Isom}(X)=\{\phi\colon X\to X\}$ be its group of isometries.
Question: Is there a point $x\in X$ fixed by all $\phi\in\...
- 13.6k
1
vote
1
answer
99
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Lower bound on a norm of $\mathbb{CP}^2$ inducing a lower bound on the Euclidean norm of $\mathbb{C}^3$
Let $|\cdot|$ denote the usual Euclidean norm on $\mathbb{C}^3$ and fix some arbitrary metric $\rho$ on $\mathbb{CP}^2$. For $\delta > 0$ and any set $\hat{P} \subset \mathbb{CP}^2$, define the $\...
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2
votes
0
answers
51
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Smallest doubling subset of a set in a metric space
Let $(X,d)$ be a separable metric space and $A\subseteq X$ be compact.
Since every finite set is doubling then, the collection $\mathcal{A}$ of doubling subsets of $A$ cannot be empty. My initial ...
- 195
1
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1
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124
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A neighborhood $Y$ of a set $X$ such that the line segment connecting any point in $Y$ and its projection to $X$ is contained in $Y$
A direct line from a point $p$ to a set $X$ is a line segment with one endpoint at $p$ and one endpoint in $X$, which is as short as any other line segment from $p$ to $X$. Given a closed set $X$ and ...
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0
answers
114
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Why does the solution to pendulum problem with the geometric approach of Jacobi metric does not correspond to the solution with Lagrangian approach? [closed]
When we solve the pendulum problem with EL equation, we get to the differential equation $\ddot{q}+\frac{g}{l}\sin q=0$
but when I apply the substitution $t \rightarrow t\sqrt\frac{g}{l}$ and ...
1
vote
0
answers
65
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Are Carnot groups ever CAT(𝜅) spaces?
Let $G$ be a free Carnot group of homogeneous dimension $d$, equipped with the Carnot–Carathéodory metric. Is $(G,d)$ ever $\operatorname{CAT}(\kappa)$ for some $\kappa\in \mathbb{R}$?
- 195
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0
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169
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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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