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

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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:(...
Keshav Srinivasan's user avatar
36 votes
2 answers
2k 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 ...
John Baez's user avatar
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35 votes
6 answers
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Trigonometry / Euclidean Geometry for natural numbers?

Let $d(a,b) = 1 - \frac{2\gcd(a,b)^3}{ab(a+b)}$ be a metric on natural numbers without $0$. The metric space $X = \{x_0,x_1,\cdots,x_n\},n>2$ is isometric embeddable in $\mathbb{R}^n$ if and only ...
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30 votes
3 answers
1k views

A funny metric over $\mathbb{N}$

$\DeclareMathOperator{\lcm}{lcm}$ Fiddling with numbers I realized that for positive integers $x$ and $y$, the quantity $$\Vert x,y \Vert=\frac{\lcm(x,y)}{\gcd(x,y)}$$ has these properties: $\Vert x,...
Luca T. Castrillón's user avatar
24 votes
8 answers
4k 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 $(M,d)$ has &...
Chill2Macht's user avatar
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24 votes
4 answers
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A reinterpretation of the $abc$ - conjecture in terms of metric spaces?

I hope it is appropriate to ask this question here: One formulation of the abc-conjecture is $$ c < \text{rad}(abc)^2$$ where $\gcd(a,b)=1$ and $c=a+b$. This is equivalent to ($a,b$ being ...
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22 votes
2 answers
2k views

Is every elementary absolute geometry Euclidean or hyperbolic?

Absolute geometry is any one that satisfies Hilbert's axioms of plane geometry without the axiom of parallels. It is well-known that it is either the Euclidean or a hyperbolic plane. For an elementary ...
Conifold's user avatar
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20 votes
2 answers
1k views

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 $y$?
Wolfgang Spindeler's user avatar
18 votes
1 answer
4k views

reference for "X compact <=> C_b(X) separable" (X metric space)

I know (and am able to prove via Stone-Čech compactification) that the following is correct: Theorem: A metric space is compact if and only if its space of bounded, continuous, real-valued ...
Wolfgang Loehr's user avatar
18 votes
3 answers
7k views

Quotient of metric spaces

Let $(X,d)$ be a compact metric space and $\sim$ an equivalence relation on $X$ such that the quotient space $X/\sim$ is Hausdorff. It is well known that in this case the quotient is metrizable. My ...
burtonpeterj's user avatar
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18 votes
1 answer
852 views

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

Can we compute the Gromov-Hausdorff distance $d(\mathbb{S}_n,\mathbb{S}_m)$ for two different spheres $\mathbb{S}_n$ and $\mathbb{S}_m$, $m\neq n$? We consider the spheres with the metrics induced by ...
Hu xiyu's user avatar
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17 votes
4 answers
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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: ...
Joseph O'Rourke's user avatar
17 votes
1 answer
354 views

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 [...
Anton Petrunin's user avatar
16 votes
5 answers
875 views

Which metric spaces have this superposition property?

Let $A \subset X$ and $B \subset X$ be two isometric subsets of a metric space $X$. So there is an isometry $f: A \to B$. Say that a metric space $X$ has the superposition property (my terminology) ...
Joseph O'Rourke's user avatar
16 votes
1 answer
588 views

If all balls around two points are isometric... -- manifold version

This question is a natural follow-up of this other question, asked earlier today by wspin. Let's say that a metric space $(X,d)$ has two poles if: there are two distinct points $x$, $y$ such that ...
Marco Golla's user avatar
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16 votes
1 answer
612 views

Does there exist a ``continuous measure'' on a metric space?

Let $X$ be a separable complete metrizable space. Does there exist a complete metric $d$ and a Borel measure $\mu$ such that (a) $\mu(B_r(x))<\infty$ for every open ball $B_r(x)$ of radius $r>...
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15 votes
3 answers
6k views

A metric for Grassmannians

I'm reading an article by Ricardo Mañé, "The Hausdorff dimension of horseshoes of diffeomorphisms of surfaces" (https://doi.org/10.1007/BF02585431). I'm having a technical problem. Sorry for ...
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15 votes
5 answers
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Converse to Banach's fixed point theorem?

Let $(X,d)$ be a metric space. Banach's fixed point theorem states that if $X$ is complete, then every contraction map $f:X\to X$ has a unique fixed point. A contraction map is a continuous map for ...
Xandi Tuni's user avatar
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15 votes
2 answers
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In infinite dimensions, is it possible that convergence of distances to a sequence always implies convergence of that sequence?

This is a cross-posted on MSE here. Let $(X,d)$ be a metric space. Say that $x_n\in X$ is a P-sequence if $\lim_{n\rightarrow\infty}d(x_n,y)$ converges for every $y\in X.$ Say that $(X,d)$ is P-...
Nikhil Sahoo's user avatar
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15 votes
1 answer
671 views

Is the topology generated by this weaker notion of a metric necessarily metrisable?

The triangle inequality seems much stronger than necessary for a lot of analysis. So I will define a "loose metric" on a set $X$ to be a function $d \colon X \times X \to [0,\infty)$ with ...
Julian Newman's user avatar
15 votes
1 answer
963 views

covering a separable metric space by small balls

Let $(X,d)$ be a separable metric space. Can $X$ always be covered by a sequence of balls $B(x_i,r_i) (i=1,2,\dots)$ s.t. radii $r_i$ tend to 0?
Fedor Petrov's user avatar
15 votes
2 answers
2k views

Where does the Lebesgue differentiation theorem fail?

The Lebesgue differentiation theorem says that for certain metric spaces $X$ (see below), any Borel measure $\mu$ that is finite on bounded sets and any $f: X \rightarrow \mathbb{R}$ locally $\mu$-...
Vanessa's user avatar
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15 votes
1 answer
1k views

In ZF, when is a disjoint union of metrizable spaces metrizable?

It is easy to see that the disjoint union $\bigsqcup_i X_i$ of a collection of metric spaces is metrizable, simply by rescaling or chopping off the individual metrics to have diameter at most one, and ...
David Feldman's user avatar
14 votes
1 answer
416 views

Does existence of midpoints imply intrinsic?

It is well-known, that a complete metric space, where any two points have a midpoints ($\forall x,y~ \exists z:~d(x,z)=d(y,z)=\frac{d(x,y)}{2}$) is strictly intrinsic, in the sense that any $x,y$ can ...
erz's user avatar
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13 votes
5 answers
1k views

A generalization of metric spaces

Let $(L,<,+)$ be a structure such that (1) $<$ is a linear order of $L$, (2) $L$ has a least element 0, (3) $+$ is a binary function on $L$ that behaves like addition of positive real numbers, i....
Monroe Eskew's user avatar
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13 votes
5 answers
4k views

Measure of the support of a Borel probability on a metric space

Does the support of a Borel probability measure always have full measure in a metric space? I know this is true for separable metric spaces, and locally compact metric spaces. Is it true in general?
Autoleech's user avatar
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13 votes
2 answers
698 views

Smooth Urysohn's lemma on Fréchet spaces

Let $V$ be a Fréchet topological vector space. Let $K_0$ and $K_1$ be two closed subsets which are disjoint. I wish to show the existence of a Fréchet-smooth function $f:V\to [0,1]$ whose restriction ...
André Henriques's user avatar
13 votes
2 answers
1k views

Baire Category Theorem for complete uniform spaces

The version of the Baire Category Theorem I have in mind is the statement that a countable intersection of dense open subsets of a complete metric space is dense. The question is: is it likewise ...
Jonathan Gleason's user avatar
13 votes
1 answer
3k views

metric on the space of real analytic functions

Hello, this question may be simple but I couldn't find a reference. Let $E$,$F$ be real Banach spaces and $\Omega\subset E$ be a bounded domain and let $C_b^{\omega}(\Omega,F)$ be the vector space of ...
Mirko's user avatar
  • 223
13 votes
1 answer
757 views

Euclidean tangent cone implies Riemannian manifold

It is known that given a Riemannian manifold, then the tangent cone (as a metric space) at any point $p$ is isometric to the tangent space at $p$, with the metric given by the metric tensor. Is ...
geodude's user avatar
  • 2,129
13 votes
1 answer
3k views

Modulus of Continuity

I originally posted this question on math.stackexchange (https://math.stackexchange.com/questions/83182/modulus-of-continuity-take-2), but it's been a few days and I haven't received any correct ...
Paul Siegel's user avatar
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13 votes
1 answer
3k views

Does this metric have an official name? Lévy metric? Ky Fan metric?

Let $X$ and $Y$ be random variables taking values in a separable metric space $(S,d)$. The metric I have in mind is $$\rho(X,Y) = \mathbb{E}[\min\{d(X,Y),1\}]$$ if $X$ and $Y$ take values in the a ...
Jason Rute's user avatar
  • 6,237
13 votes
0 answers
636 views

Covering number estimates for Hölder balls

Let $\alpha \in (0,1]$, $r>0$ and $L>0$, and positive intwgers $n$ and $m$. The Arzela-Ascoli Theorem guarantees that the set $X(\alpha,L,r)$ of $f:[-1,1]^n\rightarrow [-r,r]^m$ with $\alpha$-...
ABIM's user avatar
  • 4,989
12 votes
5 answers
1k views

Examples of metric spaces with measurable midpoints

Given a (separable complete) metric space $X=(X,d)$, let us say $X$ has the measurable (resp. continuous) midpoint property if there exists a measurable (resp. continuous) mapping $m:X \times X \to X$ ...
dohmatob's user avatar
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12 votes
1 answer
886 views

Converse to Banach’s fixed point theorem for ordered fields?

Suppose $R$ is an ordered field. Call a continuous map $f: R \rightarrow R$ a contraction if there exists $r < 1$ (in $R$) such that $|f(x)-f(y)| \leq r |x-y|$ for all $x,y \in R$ (where $|x| := \...
James Propp's user avatar
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12 votes
1 answer
562 views

Is $\ell_p$ $(1<p<\infty)$ finitely isometrically distortable?

Let $Y$ be a Banach space isomorphic to $\ell_p$, $1<p<\infty$. Is it true that any finite subset of $\ell_p$ is isometric to some finite subset of $Y$? It seems to me that it is an interesting ...
Mikhail Ostrovskii's user avatar
11 votes
3 answers
1k views

Universal sets in metric spaces

(I am cross-posting this from math.SE as it seems to be slightly over the top for that site.) I saw in the class the theorem: Suppose $X$ is a separable metric space, and $Y$ is a polish space (...
Asaf Karagila's user avatar
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11 votes
2 answers
675 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 ...
CatInTheBag's user avatar
11 votes
0 answers
515 views

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 ...
fedja's user avatar
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10 votes
4 answers
2k views

When do isometric actions exist?

Let $X$ be a metrizable topological space and $G$ be a locally compact group. Given a continuous (left) action of $G$ on $X$, is there a metric on $X$, compatible with the topology, for which the ...
Kamran Reihani's user avatar
10 votes
1 answer
549 views

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\...
M. Winter's user avatar
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10 votes
1 answer
786 views

Completely Metrizable Space and Baire Theorem

Is well know that completely metrizable spaces are Baire's spaces. Reciprocally, if $X$ is a Baire's metric space, then $X$ is completely metrizable?
Hugo Rafael Oliveira Ribeiro's user avatar
10 votes
1 answer
436 views

Probability that a random distance function is metric

Take a random $n \times n$ nonnegative symmetric matrix $D$ with zero diagonal. What is the probability that it is an abstract distance matrix, i.e. satisfies $D_{xy}+D_{yz} \geq D_{xz}$ for all index ...
Felix Goldberg's user avatar
10 votes
2 answers
427 views

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 ...
James Hanson's user avatar
  • 10.3k
10 votes
1 answer
544 views

Are packing-homogeneous spaces homogeneous?

Given a metric space (M,d) define the packing function P(x,R,r) to be the maximum number of non-intersecting balls of radius r with centers in the ball B(x,R). Let’s call M packing-homogeneous if the ...
Yevgeny Liokumovich's user avatar
10 votes
0 answers
781 views

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 ...
user avatar
9 votes
4 answers
3k views

Is the space of Radon measures a Polish space or at least separable?

Background: I work on a SPDE problem where in order to apply Prokhorov's theorem I need that some measure space is Polish space. And additionaly it would be good if that space is Banach space. Earlier ...
Mark's user avatar
  • 647
9 votes
1 answer
571 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 ...
Morteza Azad's user avatar
9 votes
3 answers
799 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 ...
Chill2Macht's user avatar
  • 2,622
9 votes
3 answers
819 views

What makes a distance?

In the answers to my previous question, I learned that there are different concepts of distance, that is of distance-like functions with the usual metric being only the most popular and important one. ...
Hans-Peter Stricker's user avatar

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