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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Differentiability of distance to a closed convex set [closed]
Let $( \mathbb{R}^d, \| \mathbf{x}\|_2 )$ be a Euclidean Space. For any nonempty closed convex set $A\subseteq \mathbb{R}^d$, we define
\begin{align}
d(\mathbf{x}, A) = \inf \{ \| \mathbf{x} - \mathbf{...
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Does there always exists a path $g:[0,1] \rightarrow X$ from $f(0)$ to $f(1)$ that has the same image as $f$ and …?
Suppose $(X,d)$ is a metric space and $f:[0,1] \rightarrow X$ is a path in $X$ with non-zero finite length $L$. Then, does there always exist a path $g:[0,1] \rightarrow X$ from $f(0)$ to $f(1)$ that ...
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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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1
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1
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Does the compactness of parameter of distribution function imply the compactness of the distribution (or probability measure) in Wasserstein space?
For a family of probability measures sharing the same form of distribution function $F(x; p)$ with different parameters (i.e., $p$'s), if the parameter falls in a compact subset of real line, can we ...
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17
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1
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Hopping geodesics
Is there a complete metric space $X$ with the following property?
For any pair of points $p,q\in X$ there is unique minimizing geodesic $[pq]_X$ that connects $p$ to $q$, but the map $(p,q)\mapsto [...
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1
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What is the quotient (pseudo)metric $d_\sim$ and how do I identify the infimum of possible sequences in this instance?
Let $Z$ be the the set of dyadic and ternary rationals in the interval $\left[\frac12,1\right)$ whose 3-adic valuation is either $-1$ or $0$, with the standard absolute value topology inherited from ...
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Reference request: extendability of Lipschitz maps as a synthetic notion of curvature bounds
In the lecture Notions of Scalar Curvature - IAS around 8:00, Gromov states the following result, which he claims he does "slightly uncarefully":
Suppose $(X,g_X)$ and $(Y,g_Y)$ are ...
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3
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1
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Expected measure of a ball in a probability space with a metric
Assume we are given a probability space $(\mathbb{X}, \mathcal{X}, \mathbb Q)$ and a measurable distance function defined on it $d:\mathbb{X}\times \mathbb{X}\to \mathbb{R}^+\cup\{0\}$ that conforms ...
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A rather non-$F_\sigma$ Borel set
I asked this question at MSE a week ago, but received no answer, so I cross-post it here.
I obtained a negative answer to this MSE question provided each metric space $X$ such that $|X|=\frak c$ 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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Are there any major differences in metric topologies and "non-symmetric" metric topologies
Let $X$ be a set and let $d:X\times X\rightarrow [0,\infty)$ satisfy all the axioms of a metric besides symmetry (i.e.: $d$ is a quasi-metric). Define a topology $\tau_{d:+}$ on $X$ induced by $d$ as ...
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Stability of Hajłasz-Sobolev class under post-composition
Informally: When is a Sobolev function, post-composed by a vector-valued function still Sobolev?
Assumptions/Setup
Let $(X,d_X,m_X)$ and $(Y,d_Y,m_Y)$ be complete and separable metric measure spaces; ...
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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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Relationship between Hausdorff convergence of sets and indicator functions
Let $\{K_n\}_n$ be a sequence of compact subsets of a metric space $X$, and $K\subset X$ be compact. If $K_n$ Hausdorff converges to $K$, i.e.:
$$
\lim\limits_{n\to\infty} d_{\mathrm H}(K_n,K) = \max\...
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Is there a theory of partially-defined metric spaces?
Is there a theory of metric spaces in which the distance between a given pair of points need not be defined?
I'm aware that there is a theory of partial metric spaces, but these deal with a different ...
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1
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1
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Inferring the modulus of continuity
Let $f:X\rightarrow Y$, $g:Y\rightarrow Z$ be uniformly continuous functions between metric spaces $X,Y,Z$ with moduli of continuity $\omega_f$ and $\omega_g$, respectively. Suppose that we know that ...
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Relationship between Hausdorff dimension and covering number
Let $(X,d)$ be a compact metric space and recall that the $\epsilon$-external covering number $\mathcal{N}^{\epsilon}(X)$ of $X$ is defined by:
$$
\mathcal{N}^{\epsilon}(X) := \inf\left\{
N\in \mathbb{...
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2
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If all balls at $x$ and $y$ are isometric is there an isometry sending $x$ to $y$?
Let $(X,d)$ be a metric space and $x,y \in X$. Assume that for all $r > 0$ the balls $B_r(x)$ and $B_r(y)$ are isometric.
Is it true that there exists an isometry of $X$ sending $x$ to $y$?
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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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Lipschitz-free space of countable uniformly discrete metric space
I assume here that the reader is familiar with the concept of Lipschitz-free space $\mathcal{F}(X)$ of a metric space $X$. I will follow the definition of $\mathcal{F}(X)$ as the completion of the ...
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2
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Which points in the Samuel compactification of a metric space $X$ are limits of uniformly discrete subsets of $X$?
Given a metric space $(X.d)$ the Samuel compactification of $X$, written $sX$, is the unique compactification with the property that if $Y$ is an arbitrary compact Hausdorff space and $f:X\rightarrow ...
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Topological properties inherited by the Hausdorff metric space
Let $(X,d)$ be a metric space and $(K_X , h_d)$ be the associated metric space of nonempty compact subsets of $X$ with the Hausdorff metric. It is well known that $K_X$ inherits certain topological (...
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Does uniform continuity of bounded continuous functions implies the same for all continuous functions on a uniform space?
Recently I came to know about Atsuji space from the paper1. A metric space $X$ is called an Atsuji space if every real-valued continuous function on $X$ is uniformly continuous. Strikingly I have ...
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Is every Baire metric space a complete metric space in disguise?
I am currently giving lectures in real analysis and a student asked an interesting question I couldn't answer, so I'm posting it here:
Let's say that a metric space $X$ is Baire if every countable ...
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1
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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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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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1
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Uniform distance from a discontinuous function is continuous
Define the metric $d(f,g)\triangleq \sup_{x \in [0,1]} \|f(x)-g(x)\|$ on the set $\operatorname{B}$ of uniformly bounded functions from the interval $[0,1]$ to $\mathbb{R}$, fix $g \in \operatorname{B}...
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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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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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4
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Metrics for lines in $\mathbb{R}^3$?
I seek a metric $d(\cdot,\cdot)$ between pairs of (infinite) lines in $\mathbb{R}^3$.
Let $s$ be the minimum distance between a pair of lines $L_1$ and $L_2$.
Ideally, I would like these properties:
...
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4
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"Snowflaked" Hausdorff metric
Let $(X,d_X)$ be a compact metric space and let $Comp(X)$ be the set of closed subsets of $X$ with the Hausdorff metric:
$$
D(A,B)\overset{\text{def}}{=} \, \max\left\{\sup_{b\in B}\,d_{A}(b),\sup_{a\...
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Gromov-Hausdorff distance between weighted tree graphs
I would like to measure the similarity between a pair of weighted tree graphs. According to this post, this can be done by regarding the trees as metric spaces and then applying the Gromov-Hausdorff ...
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2
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Balls in Lawvere metric spaces
Let $V$ be the monoidal category $[0,\infty)$ (as a poset) with $+$ and $0$. Lawvere shows that $V$-enriched categories are a more natural generalisation of the notion of a metric space (note no ...
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Is there a name for this geometric property of metric spaces?
My research has lead me to metric spaces $(M, \rho)$ which have the following geometric property:
Suppose $x, y \in M$ and $r, s > 0$ such that
$(x, r) \neq (y, s)$,
$B[y; s] \subseteq B[x; r]$,
$...
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0
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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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Domains in $\mathbb{R}^n$ for which Hajlasz-Sobolev spaces and Sobolev Spaces are the same
I'm reading Heinonen's book on metric measure spaces. He writes that for general domains $\Omega \subset \mathbb{R}^n$, $M^{1,p}(\Omega) \subset W^{1,p}(\Omega)$ where the former are Hajlasz-Sobolev ...
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Continuous extension preserving modulus of continuity
Let $X$ be a (non-empty) compact subset of $D(0,M):=\left\{x\in \mathbb{R}^n:\, \|x\|\leq M\right\}$, and let $f:X\rightarrow Y$ be uniformly continuous; for some metric space $Y$. Are there any ...
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0
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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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Condition for: A simple quotient metric induced by surjective map + equivalence relation
Let $X$ be a metric space and let $f:X\rightarrow Z$ be a surjective map onto some set $Z$. Define the pseudo-metric $d_f$ on $Z$ by:
$$
d_f(z_1,z_2)\triangleq \inf_{\underset{f(x_i)=z_i}{x_i\in X}}
\...
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Two questions around the $abc$-conjecture
Let $d(a,b) = 1-\frac{2 \gcd(a,b)}{a+b}$, $d_{ABC}(a,b) = 1-\frac{2\gcd(a,b)^3}{ab(a+b)}$ be two metrics on natural numbers.
The abc-conjecture can be formulated using these two metrics as:
For ...
user6671
9
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1
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On the Large Cardinal Strength of Normal Moore Space Conjecture
In his seminal 1937 paper, Jones [1] proved the following result about Moore spaces:
Theorem. (Jones) If $2^{\aleph_0}<2^{\aleph_1}$ then all separable normal Moore spaces are metrizable.
Then ...
5
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1
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415
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Spreading $n$ points in $\{0,1\}^n$ as far as possible
Given a positive integer $n$, the Hamming distance $d^H_n(x,y)$ of $x,y\in \{0,1\}^n$ is defined by $$d^H_n(x,y) = |\{k\in\{0,\ldots,n-1\}: x(k)\neq y(k)\}|.$$
We say that a positive integer $s$ is $...
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Distance between two metric spaces
I am given two metric spaces as two arrays of the same size. Each one is supposed to represent distance between vertices on a mesh in R^3. The meshes are assumed to have the same number of vertices ...
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1
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0
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70
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Injectivity of post-composition operator
Let $X$, $Y_1,Y_2$, and $Z$ be separable metric spaces. Let $C(X,Y)$ be the topological space of continuous functions from $X$ to $Y$ equipped with its compact-open topologies. Fix a continuous ...
8
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1
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Intersection of nested open ball in complete metric spaces is nonempty?
My question is that whether the following statement is true or not.
In a complete metric space $(X, d)$, if a sequence of open balls $\{B(x_i, r_i)\}_{i=1}^\infty$ satisfies
$$
\exists \epsilon > ...
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2
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264
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Existence of a Hölder-free space
The Lipschitz-free or Arens-Eells space over a pointed separable metric space $(X,0,d)$ is a well-studied object. My question is, is an analogos Hölder-free space; for a fixed Hölder constant $\alpha&...
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0
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69
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Holder-continuous barycenter maps
Let $(X,d)$ be a complete locally-compact metric space. We define the $p$-barycenter map as a continuous function:
$$
\beta:\mathcal{P}_p(X)\rightarrow X,
$$
which is a right-inverse of the map ...
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3
answers
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Is the hyperspace of the Hilbert cube homeomorphic to the Hilbert cube
Question: Is the hyperspace of the Hilbert cube $H=[0,1]^\mathbb {N}$ homeomorphic to $H$?
Remarks and definitions:
1) The Hilbert cube $H$ is a compact metric space, where the metric is given by ...
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0
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Conditions under which a metric on a Riemannian manifold is induced by a Riemannian metric
Let $(M, g)$ be a Riemannian manifold. Lately, I've grown interested in what you may call a "modified geodesic" problem. Given some smooth, non-negative scalar field $V$ on $M$ (aptly called ...
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3
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1
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233
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Show identity for a norm on Fréchet differentiable functions on a Banach space
Let $E$ be a $\mathbb R$-Banach space, $v:E\to(0,\infty)$ be continuous with $$\inf_{x\in E}v(x)>0\tag1,$$ $r\in(0,1]$ and$^1$ $$\rho(x,y):=\inf_{\substack{c\:\in\:C^1([0,\:1],\:E)\\ c(0)=x\\ c(1)=...
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