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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) $d(x,y)+d(y,z) \geq d(x,z)$.

4
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1answer
297 views

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 $...
25
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3answers
579 views

What is the structure preserved by strong equivalence of metrics?

Let $X$ be a set. Then we can define at least three equivalence relations on the set of metrics on $X$. We say that two metrics $d_1$ and $d_2$ are topologically equivalent if the identity maps $i:(...
2
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1answer
58 views

Stretching map of $n$ points from $\{0,1\}^n$ to $\{0,1\}^{n+1}$ with respect to their Hamming distance

Given a positive integer $n$, the Hamming distance $d^H_n(x,y)$ of $x,y\in \{0,1\}^n$ is defined by $$d^H_n(x,y) = |\{k\in\{0,\ldots,n-1\}: x(k)\neq y(k)\}|.$$ Given an integer $n>0$ and a set $S\...
1
vote
1answer
95 views

Is every finite graph isomorphic to the proximity graph of some $S\subseteq \mathbb{R}^n$?

This is the question that I should have asked before asking this older question. If $(X,d)$ is a metric space, we associate with it a simple, undirected graph, called its proximity graph $G(X,d)$ ...
2
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1answer
69 views

Spatial dimension of a finite graph

If $(X,d)$ is a metric space, we associate with it a simple, undirected graph, called its proximity graph $G(X,d)$ given by $V(G(X,d)) = X$ and $$E(G(X,d)) = \big\{\{x,y\}:x\neq y\in X \text{ and } d(...
1
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1answer
62 views

Takahashi minimization theorem for lower pseudo-continuous functions on complete metric spaces

Takahashi minimization theorem says : Let $(X,d)$ is a complete metric space, $f:X\to \mathbb{R}\cup\{+\infty\}$ is a proper(not constantly +$\infty$) lower semi continuous function, which is bounded ...
3
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1answer
148 views

Greedy simplices in an ultrametric space (generalized Bhargava $p$-orderings)

Let $\left(U, d\right)$ be a finite ultrametric space -- that is, $U$ is a finite set, and $d : U \times U \to \mathbb{R}_{\geq 0}$ is a metric on $U$ such that every $x, y, z \in U$ satisfy $d\left(x,...
0
votes
1answer
92 views

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 ...
4
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2answers
148 views

largest diameter of intersection of two balls

Two closed balls with a common radius are positioned so that the centre of either ball is on the boundary of the other. I am interested in the extremal diameter of their intersection, in an arbitrary ...
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0answers
29 views

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 ...
0
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1answer
67 views

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 $...
1
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2answers
98 views

Extending homeomorphisms between compact metric subsets

Let $X$ be a compact metric, second countable space with finite covering dimension. Let $A,B$ be two closed subsets of $X$. Assume that $h:A\to B$ is a homeomorphism. Is it possible to extend $h$ to a ...
8
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1answer
272 views

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 > ...
0
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1answer
77 views

Measure on group invariant under group action on metric space

This is a question very similar to I recently asked on mathexchange, but different enough to get its own entry in MO. The setting is still the same. I consider the metric space $\mathbb{R}$ and the ...
1
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1answer
66 views

A property on some unbounded metric spaces

Suppose that $(X,d)$ is a metric spaces. Which condition(s) can guaranties the following property: $\forall x, \forall y \in X, \exists \{z_n\}$ such that $\lim_{n\to +\infty } d(x,z_n)=+\infty$ and $\...
3
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0answers
60 views

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 paper. A metric space $X$ is called an Atsuji space if every real-valued continuous function on $X$ is uniformly continuous. Strikingly I have found ...
9
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2answers
253 views

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 ...
8
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0answers
197 views

Universally meager spaces and large cardinals

Definition: (Todorcevic) A subset $A$ of a topological space $X$ is called universally meager if for every Baire space $Y$ and every continuous $f : Y \to X$ which is nowhere constant (not constant on ...
2
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1answer
115 views

Path Metric Topology

Is there an example of a metric space $(X,d)$ whose corresponding path metric, $d^\prime$ generates a strictly finer topology compared to the topology generated by $d$?
9
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1answer
402 views

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 ...
3
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1answer
125 views

Quotient of compact metrizable space in Hausdorff space

Lets $X$ be a compact metrizable space and $f:X\to Y$ be a quotient map such that $Y$ equipped with the quotient topology is Hausdorff. Thus $Y$ is metrizable. Lets $\sim$ be an equivalence relation ...
3
votes
1answer
70 views

Are there any statistical metrics that satisfy this kind of condition?

Let $f=N(\mu,\sigma^2)$ be a univariate normal distribution with mean $\mu$ and variance $\sigma^2$ and let $f_1 = N(\mu+\epsilon,\sigma^2)$ and $f_2=N(\mu,(\sigma+\epsilon)^2)$ be some small ...
2
votes
1answer
116 views

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 ...
3
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1answer
265 views

Earth movers distance (EMD) between two multivariate normals. Is it negative definite distance?

I was looking at the closed form formula for 2-Wassersteins distance for multivariate normal distribution on Wikipedia. https://en.wikipedia.org/wiki/Wasserstein_metric#Normal_distributions It has a ...
0
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1answer
53 views

Is the function $g$ always injective where $g$ is obtained by lipschitz re-parametrization

Suppose $(X,d)$ is a metric space with the nearest point property and $a,b \in X$ with $a \ne b$. Suppose there is a path of finite length in $X$ from $a$ to $b$ and let $m$ be the infimum of the ...
0
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1answer
131 views

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 no-zero finite length $L$. Then, does there always exists a path $g:[0,1] \rightarrow X$ from $f(0)$ to $f(1)$ that ...
7
votes
1answer
402 views

Totally bounded spaces and axiom of choice

Wikipedia article on totally bounded spaces states "... the completion of a totally bounded space might not be compact in the absence of choice." Where is the axiom of choice used, and do you need it ...
4
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0answers
317 views

Does rate of convergence in probability come from a metric?

In general, when we talk about convergence of a sequence, we need a topological space. If we want to talk about a rate of convergence, we need to quantify how far away one element of the sequence is ...
0
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1answer
58 views

Hausdorff convergence of preimages of discrete-valued functions

Suppose $f_n$, $f:X\to K$ where $K$ is a finite set and $(X,d)$ is a metric space. Suppose also that $f_n(x)\to f(x)$ for all $x\in X$ (pointwise convergence). Finally, let $d_H$ be the Hausdorff ...
1
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1answer
347 views

Base of topology for metric-like space

Let $X$ be a nonempty set and $p:X\times X\rightarrow\mathbb{R}^+ $ be a function satisfying the following conditions for all $x,y,z\in X$: \begin{align} &1)\enspace p(x,y)=0\implies x=y \\ &2)...
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0answers
132 views

Gromov-Hausdorff relative compactness without curvature restrictions

A famous theorem of Gromov says that the set of compact Riemannian manifolds with $Ric \geq c$ and $\text{diam} \leq D$ is relatively compact in the Gromov-Hausdorff metric. Chapter 10 of the book by ...
33
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2answers
1k views

Is there a “universal” connected compact metric space?

Fact 1. The Cantor set $K$ is "universal" among nonempty compact metric spaces in the following sense: given any nonempty compact metric space $X$, there exists a continuous surjection $f\colon K \to ...
3
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0answers
151 views

Graph contained in a metric space

I have a metric space $X$ and a graph $G=(V,E)$ whose set of vertices is a subset $V\subset X$ (and $E$ is the set of edges, which is a symmetric subset of $V\times V$). For each $v\in V$, the set of ...
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0answers
171 views

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: ...
0
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1answer
232 views

$\ell_1$ and $\ell_\infty$ as complementary subspaces of Banach space

Let $X$ be a Banach space, and let $X'\subset X$ - its subspace. Then the following propositions are true: $X'$ is closed, $X/X' \cong \ell_1 \Rightarrow X'$ is complementary; $X' \cong \ell_\infty ...
14
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0answers
395 views

How to compute the Gromov-Hausdorff distance between spheres $S_n$ and $S_m$?

There is the question, because when we consider the Gromov-Hausdorff distance, we must fix the metric, so we use the natural metric induced from the embedding $\mathbb{S}_n \to \mathbb{R}^{n+1}$. Is ...
2
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0answers
50 views

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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0answers
141 views

A characterization of Cauchy filters on countable metric spaces?

Given a filter $\mathcal F$ on a countable set $X$, consider the family $$\mathcal F^+:=\{A\subset X:\forall F\in\mathcal F\;(A\cap F\ne\emptyset)\}.$$ The following characterization is well-known. ...
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0answers
59 views

Given a topological field with non-trivial topology given by a metric, is the metric necessarily translation-invariant? [closed]

If the topology is locally compact and non-trivial, then the metric is usually translation-invariant or even arises from a norm. But is it necessary to be invariant? I failed to give a counterexample.
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0answers
34 views

Equivalent of linearity for the Fréchet mean under translations

Following Pennec (2006), the Fréchet mean for a set of draws $\{x_i\}_{i=1,\dots,n}$ from a metric space $M$ with distance function $d$ is defined as the minimiser of the variance of the draws, $$ E[\...
2
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1answer
169 views

Ultraproduct of metric spaces

Let $I$ be a set and $\mathcal{U}$ be an ultrafilter on $I$. Suppose that $(X_{i}, d_{i})_{i\in I}$ is a family of pointed metric spaces with a distinguished point $e_{i}$ for each $i\in I$. We set $$...
0
votes
1answer
207 views

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) ...
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2answers
211 views
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0answers
158 views

Results that are easier in a metric space

Are there any significant results in the theory of metric spaces that (are considerably more difficult to reproduce/have not been reproduced) in the theory of uniform spaces? In particular, I'm ...
19
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2answers
592 views

Why doesn't this construction of the tangent space work for non-Riemannian metric manifolds?

In the 1957 paper, On the differentiability of isometries, Richard S. Palais gives a way to construct the tangent spaces of a Riemannian manifold using only its metric space structure (Theorem, p.1). ...
9
votes
3answers
446 views

When is “metric dimension” well defined?

A subset $B$ of a metric space $(M,d)$ is called a metric generating set if and only if $$[\forall b \in B, d(x,b)=d(y,b)] \implies x = y \,. $$ A metric generating set $B$ is called a metric basis ...
20
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7answers
2k views

When does a metric space have “infinite metric dimension”? (Definition of metric dimension)

Definition 1 A subset $B$ of a metric space $(M,d)$ is called a metric basis for $M$ if and only if $$[\forall b \in B,\,d(x,b)=d(y,b)] \implies x = y \,.$$ Definition 2 A metric space $(M,d)$ ...
1
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1answer
43 views

Cauchy subsequences in “Hausdorff Cauchy sets”

This is a follow-up to an older question. Let $(X,d)$ be a metric space and let $(A_n)_{n\in\mathbb{N}}\subseteq X$ be a sequence of bounded non-empty subsets of $X$ such that for all $\varepsilon &...
7
votes
1answer
334 views

Hausdorff distance and Cauchy sequences

This is a generalization of an older question. Let $(X,d)$ be a metric space and let $(A_n)_{n\in\mathbb{N}}\subseteq X$ be a sequence of non-empty closed subsets such that for all $\varepsilon > ...
0
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0answers
65 views

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, ...