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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Li-Yorke chaos: the non compact case
1) Is there any notion of Li-Yorke chaos for non compact (metric) spaces $X$ and non continuous transformation $f:X \rightarrow X$? Could you bring some references?
2) I mean, why are so important ...
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Connected metric spaces without connected separable subspaces
My question is the following:
Does there exist a connected metric space $\ X,\ $ where $\ |X|>1,\ $ which contains no separable connected subspace $\ Y\ $ with $\ |Y|>1\ $?
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Upper bound on the Levy-Prokhorov distance between the distributions of continuous Gaussian processes in terms of their covariances
Denote by $d$ the supremum metric on the space $C[0,T]$ of continuous real-valued functions on $[0,T]$:
$$
d(f,g) = \sup_{t \in [0,T]} |f(t)-g(t)|.
$$
Let $\rho$ be the Levy-Prokhorov metric on the ...
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Complexity for determining whether a given metric space is hyperconvex?
Suppose I am given a finite metric space as a distance matrix. What is the complexity of determining whether this metric space is hyperconvex?
Definition: A metric space is said to be hyperconvex if ...
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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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Metric projection on closed convex sets in Busemann space
I am looking for a proof of the following statement:
Let $X$ be a complete Busemann space. For any point $x\in X$ and any nonempty closed convex set $A\subseteq X$, there is a unique $a\in A$ such ...
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Proving that family of sets has non-empty intersection
Let's say I have an object which can be viewed as family of sets $\mathfrak{S} \subseteq 2^S$, and I want to prove that its intersection is non-empty. What is known already:
$S$ is set of measurable ...
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Isometry between punctured sphere and punctured triangle?
Setup:
Let $C_n$ be a closed $n$-simplex in $\mathbb{R}^n$ and let $r \in (0,R)$ where $R$ is the distance any one of the vertices $\{v_1,\cdots , v_{n+1}\}$ of $C_n$ to the centroid $\frac{v_1+ \...
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Conditions for a set being closed under taking complement of a ball twice
Given a subset $S$ of a finite metric space $F$ with a distance function $d(,)$ and a number $\delta > 0$ let $N_\delta(S) = \{x \in F| d(x,S)\ge \delta\}$. Is there a characterization of ...
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If $(\mathbb M, \tau)$ is a topological monoid, is $\tau$ always induced by a [left] subinvariant semimetric?
Let me start by recalling some basic definitions (just for the sake of avoiding misunderstandings due to the vocabulary of the post).
Basically following some ideas of W. Lawvere (but not his ...
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Skorokhod-like construction for sequences of random probability measures
Let $(X_i)$ be a sequence of i.i.d. random vectors with distribution $P$ on $[0,1]^d$.
Let $D \equiv D([0,1]^d)$ be the multivariate Skorokhod space, equiped with a metric $d$ that makes it Polish. ...
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Lipschitz maps with Hölder inverse preserve the doubling property
Let $K$ be a compact doubling metric space, $X$ be a metric space and $f:K\rightarrow X$ be Lipschitz with $\alpha$-Hölder inverse, where $0<\alpha<1$. Does $f(K)$ need to be doubling?
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Is there a theory of partially-defined metric spaces?
Is there a theory of metric spaces in which the distance between a given pair of points need not be defined?
I'm aware that there is a theory of partial metric spaces, but these deal with a different ...
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For Polish $X,Y$, $L^p(X,Y)$ is separable
Let $X$ and $Y$ be Polish spaces. Equip $X$ with a Borel probability measure $\mu_X$ and $Y$ with a metric $d_Y$. We can define the $L^p$ space as follows:
Definition. Define
$\begin{align}L^p(X,Y) = \...
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Statistical invariants of Riemannian manifolds
$\DeclareMathOperator\diam{diam}\DeclareMathOperator\rad{rad}\DeclareMathOperator\iso{iso}\DeclareMathOperator\com{com}\DeclareMathOperator\con{con}$A cheap way of defining invariants of Riemannian ...
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Can we control the Wasserstein metric between $\mu$ and $\nu$ by their moment difference?
Fix $p \in [1, \infty)$. Let $(\mathcal P_p(\mathbb R^d), W_p)$ be the Wasserstein space of all Borel probability measures on $\mathbb R^d$ with finite $p$-th moment. Let $D_p$ be the collection of ...
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Best way to hang a lampshade
I have a lampshade which looks like a demi-sphere but with irregular border. The goal is to hang it to the ceiling so that the border looks as horizontal as possible.
In order to formalize this, let ...
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The world of non-weak*-topologies on $\mathcal{P}(X)$
Let $X$ be a metrizable space and consider $\mathcal{P}(X)$, the set of all probability measures on $X$.
Typically, the weak*-topology is considered on $\mathcal{P}(X)$, which is a very natural ...
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Maximal cliques in neighborhood graphs of partial $k$-trees (bounded treewidth)
Background
My question is about a generalization of the following situation:
Let $M$ be a finite metric space. Given $r>0$, the $r$-neighborhood graph $N(M)_r$ has vertex set $M$ and an edge $\{x,y\...
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Is there a literature name for this concept of a "graded metric"?
Given a space $X$, I have been thinking about a function $d\colon X \times X \times \mathbb{N} \to \mathbb{R}_{\geq 0}$ (i.e. with values that are nonnegative reals) with the properties below. One may ...
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Regularity of $\sigma$-finite measure pushforwarded by completion
Let $(X, d)$ be a metric space. Let $\mu$ be a $\sigma$-finite measure defined on borel subsets of $X$. Let $i \colon X \to \hat{X}$ be an isometry on image, where $\hat{X}$ is a complete metric space ...
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Chess pieces metrics in higher dimensions
A couple of days ago, I was thinking about applying the knight (the well-known piece of chess) metric to any cubic lattice $\mathbb{N}^k$, $k \in \mathbb{N}-\{0,1\}$.
I suddenly realized that, from $k ...
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A version of Portmanteau theorem where $(\mu_n)_{n\in \mathbb N}$ is replaced by a net $(\mu_d)_{d\in D}$
Let $(E, d)$ be a metric space, $\mathcal C_b(E)$ the space of all real-valued bounded continuous functions on $E$, and $\mathcal P(E)$ the space of all Borel probability measures on $E$. For $f \in \...
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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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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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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 ...
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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 ...
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Estimating the largest radius making each ball in a finite metric space into a tree
Motivation:
Let $n$ be a positive integer and $(X,d)$ be an $n$-point metric space. Clearly, $(X,d)$ need not be a metric tree (e.g. take for example the discrete metric on $\{0,1,2\}$.
Conversely, ...
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Finite approximations to the Kuratowski/Fréchet embedding
Let $(X,d)$ be a compact doubling metric space with doubling constant $C>0$. Let $\{\mathbb{X}_n\}_{n=0}^{\infty}$ be a sequences of finite subsets of $X$ with
$$
\left\{B\left(x_k,\frac1{n}\right)...
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Does a lifted functor on $\mathbf{1Met}$ preserve isometries?
Let $\mathbf{1Met}$ denote the category of metric spaces with distance bounded by $1$ and nonexpansive maps ($1$-Lipschitz functions). I call isometry a distance-preserving map (some people require it ...
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Perturbing the approximation property from the Lipschitz-free space to stay in the Wasserstein space
Let $(X,d,x)$ be a separable pointed metric space and let $\mathcal{F}(X)$ be its Arens-Eells (also called its Lipschitz-Free space; in the case where $X$ is Banach) space. We view the $1$-...
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A generalization of metrics taking values in partial orders
I'm investigating the origin of the following notion:
Let $S=(S, +, <, 0)$ be a partially ordered semigroup with minimum $0$ (such that $<$ is invariant by the action of $+$ on both sides).
A $S$...
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What is the relationship between barycenters in the Arens-Eells sense and barycenters in the optimal transport sense
Setup:
Let $X$ be a complete pointed metric space.
Let us briefly recall that the Wasserstein space $W_1(X)$ is identifiable with a subset of the Arens-Eells (or Lipschitz-Free) space $\operatorname{...
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Almost Lipschitz embedding of compact metric measure spaces into Euclidean spaces
Let $(X,d)$ be a compact metric space, $m$ be a metric outer-measure on $X$. Are there 'mild conditions' on $X$ ensuring the existence of a positive integer $N\geq 3$ such that there exist $x_1,\dots,...
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Relationship between Hausdorff dimension and covering number
Let $(X,d)$ be a compact metric space and recall that the $\epsilon$-external covering number $\mathcal{N}^{\epsilon}(X)$ of $X$ is defined by:
$$
\mathcal{N}^{\epsilon}(X) := \inf\left\{
N\in \mathbb{...
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Closure of finite support measures in the Wasserstein metric
This is a follow-up question to this question:
Closure Wasserstein for pointmasses
Let $(X,d)$ be a metric space, and let $W_1(X)$ be the space of probability measures $\mu$ on $X$ having finite first ...
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Simplify Kantorovich–Rubinstein duality when distributions share a common marginal
Consider the product of two metric spaces $X\times Y$, and two probability distributions $\mu$ and $\nu$ on this product space. By the Kantorovich-Rubinstein duality, I can write the Wasserstein-1-...
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Examples of doubling metric spaces
I keep reading a lot of metric space results which are frames for doubling metric spaces. However, besides some obvious examples (such as Euclidean case, discrete spaces, or quasi-symmetric images of ...
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Generalized Hardy operator and Lorentz gamma spaces
I would like to find an inequality which would 'place' the generalized Hardy operator $\int_0^th(y)dy\int_y^tk^*(s)ds$ in between two Lorentz gamma spaces.
Any literature or ideas would be greatly ...
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Conditions on the inequality with a gauge norm
Let $\Phi(x)=\int_0^x \phi(y)\,dy$, $x \in \mathbb{R}_+$, be an N-function, and let $u$ be locally inferable on $\mathbb{R}_+$. Consider the gauge norm
$$
\rho_{\Phi,u}(f)=\inf\{\lambda>0: \int_{\...
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The contraction principle in quasi metric spaces
I am researching contractive mappings and I need the article of I. A. Bakhtin "The contraction principle in quasi metric spaces"(1989) or at least part where explanation is given for ...
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Weak convexity in graphs
I asked the following question in MathStackExchange, but I did not get any answer, and I think that this might be the appropriate venue for the question.
As we know, a finite undirected graph ...
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A Riemannian metric on the plane such that the intersection of every two discs is a disc, again
Is there a Riemannian metric on $\mathbb{R}^2$ (or a $2$ dimensional manifold) such that the intersection of every two open discs is an open disc, again?
As linear version of this question we ask:
...
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Existence and uniqueness of fixed point in generalized condition of triangular norm
Definition 1) A Menger space is defined as a triple $\left( S,F,T \right)$ where $S$ is a set , $F$ is a collection of distribution functions and $T$ is a triangular norm function
$T:[ 0,1 ]\...
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Generalize upper semicontinuous regularization using Borel Hierachy
Let $X$ be a metric space. Suppose a real-valued function $f:X\rightarrow \mathbb{R}$ is upper semicontinuous class $2$ if for all $c \in \mathbb{R},$ its preimage $f^{-1}(-\infty,c)$ is $F_{\sigma}$.
...
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518
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When will the upper regularization of a bounded function not defined?
Suppose $E$ is a compact metric space.
A function $f :E \rightarrow \mathbb{R}$ is upper semicontinous if for all $c \in \mathbb{R}$, $f^{-1}(-\infty, c)$ is open in $E.$
For any real-valued ...
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Equicontinuity of $\{f_{2n}\circ f_{2n-1}\}$
Let $(X,D)$ be a compact metric space and $\{f_n\}_{n\in\mathbb{N}}$ be a sequence of homeomorphisms of $(X,d)$. It is easy to see that if $\{f_n\}$ is uniformly convergent then $\{g_n\}$ defined by $...
2
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126
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Nearly injective Banach spaces
There was a problem about nearly injective metric spaces posed by Aronszajn and Panitchpakdi which I actually solved in the past but it still remains open (as long as I know) for the Banach spaces--so ...
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95
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Cover a set with balls centered at smooth functions (Ascoli theorem)
Assume $M$ to be a compact $n$-dimensional manifold, endowed with a complete metric.
Let us consider the space $C^\infty(M)$ endowed with the standard $C^\infty$ topology, i.e. generated by the ...
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First-countable topological monoids without local absorbing elements whose topology is induced by a semimetric
This is a follow up of Question 163246. For the reader's convenience, let me first copy&paste some basic definitions.
We let a semimetric on a set $X$ be a function $d: X \times X \to [0,\infty]$ ...
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