All Questions
Tagged with metric-spaces mg.metric-geometry
159 questions
49
votes
3
answers
3k
views
What happens if you strip everything but the “between” relation in metric spaces
Given a metric space $(X,d)$ and three points $x,y,z$ in $X$, say that $y$ is between $x$ and $z$ if $d(x,z) = d(x,y) + d(y,z)$, and write $[x,z]$ for the set of points between $x$ and $z$.
Obviously,...
- 887
38
votes
3
answers
3k
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:(...
- 4,597
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 &...
- 2,680
24
votes
4
answers
2k
views
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 ...
user6671
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 ...
- 1,731
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 ...
- 888
18
votes
1
answer
901
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 ...
- 697
17
votes
4
answers
2k
views
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:
...
- 151k
17
votes
1
answer
363
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 [...
- 45k
16
votes
5
answers
903
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) ...
- 151k
15
votes
3
answers
7k
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 ...
user11178
14
votes
1
answer
453
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 ...
- 5,529
13
votes
1
answer
844
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 ...
- 2,129
13
votes
0
answers
818
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 $\alpha$-...
- 5,405
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 $m:X \times X \to X$ ...
- 6,853
12
votes
1
answer
575
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 ...
- 4,878
11
votes
2
answers
722
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 ...
- 111
10
votes
1
answer
561
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\...
- 13.6k
10
votes
1
answer
560
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 ...
9
votes
3
answers
818
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 ...
- 2,680
9
votes
3
answers
836
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.
...
- 9,720
9
votes
1
answer
2k
views
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{...
- 1,127
9
votes
0
answers
489
views
Category of metric spaces
Is there a standard/good reference text that does category of metric spaces?
Say, it seems that by looking at this category one can recover everything about particular metric space up to scaling --- ...
- 45k
8
votes
1
answer
432
views
What should a meaningful notion of curvature satisfy, in the absence of a smooth structure?
There are many generalizations of various curvatures to non-smooth metric spaces (e.g. Ollivier's Ricci curvature). Suppose I have a metric space $(X,d)$ and I want to define a notion of curvature ...
- 232
8
votes
1
answer
2k
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 > ...
- 203
8
votes
1
answer
530
views
Whitney's approximation theorem for Lipschitz manifolds
In the smooth setting, Whitney's approximation theorem says the following: If $M,N$ are smooth manifolds and $f,g:M\to N$ are smooth functions that are continuously homotopic (ie there is a continuous ...
- 1,149
8
votes
1
answer
881
views
Gromov-Hausdorff convergence for non-compact metric spaces
Let $(X_i,p_i)$, $(X,p)$ be pointed connected proper metric spaces (i.e. the closures of balls are compact). Are the following two statements equivalent?
$\forall r > 0: \bar{B}_r(p_i) \stackrel{...
- 571
8
votes
0
answers
149
views
Do the $\ell^{\infty}$ and $\ell^1$ norms yield minimal doubling constants amongst all norms on $\mathbb{R}^n$?
Setting:
Let $X:=\mathbb{R}^n$ for some positive integer $n$. For each $1\le p\le \infty$ let $d_p$ denote the metric induced by the $\ell^p_n$ norm thereon.
Note that, the doubling constant of a ...
- 5,405
7
votes
1
answer
590
views
When is a metric space a snowflake?
Let $(X,d)$ be a metric space. For any $0<\epsilon<1$, we call the metric space $(X,d^{\epsilon})$; where $d^{\epsilon}(x,y)\triangleq (d(x,y))^{\epsilon}$ the $\epsilon$-snowflake of $(X,d)$.
...
7
votes
1
answer
195
views
Does there exist a countable metric space which is Lipschitz universal for all countable metric spaces?
Is there a countable metric space $U$ such that any countable metric space is bi-Lipschitz equivalent to a subset of $U$? How about $c_{00}(\mathbb{Q})$ where $\mathbb{Q}$ is the rational numbers? ...
- 73
7
votes
0
answers
493
views
A locally compact, complete metric space in which the closure of open balls coincide with the closed ball is Heine-Borel
I saw the following result stated without a proof in a paper about the isometry group of metric measure spaces:
Let $X$ be a locally compact, complete metric space such that for all $x \in X$ and $R &...
- 99
6
votes
2
answers
381
views
Sources for Alexandrov surfaces
There are two distinct notions in differential geometry associated
with A. D. Alexandrov: (1) Alexandrov spaces of courvature bounded
from below; (2) Alexandrov surfaces of bounded total curvature (...
- 16.6k
6
votes
2
answers
297
views
Can every set of points with rational distance squares be isometrically embedded in $\Bbb Q^d$?
Suppose we are given a finite family of points $p_1,...,p_n\in \Bbb R^d$, so that any two points have a rational distance square, that is,
$$\|p_i-p_j\|^2\in\Bbb Q,\quad\text{for all $i,j\in\{1,...,n\}...
- 13.6k
6
votes
1
answer
333
views
Trasportation metric (AKA Earth-Mover's, Wasserstein, etc.) as "natural" / "induced"?
Context: Given a discrete finite metric space $X$ (in my case X={0,1}$^n$ with the Hamming/L$_1$ distance), I need to define the natural or canonical metric on the set of all probability distributions ...
- 111
6
votes
1
answer
551
views
Relationship between doubling constant of a metric space and of a metric measure space
Let $(X,d,m)$ be a metric measure space. We say that it is doubling in the sense of metric spaces if for every:
$x\in X$ and every $r>0$ there exists some (metric) doubling constant $C_d\geq 0$ ...
- 195
6
votes
1
answer
348
views
Reference: Hajlasz-Sobolev Spaces with Values in a Metric Space
Let $(X,d,\mu)$ be a separable metric measure space on which every ball has positive but finite measure.
I've come across the definition of a homogeneous Fractional Hajlasz-Sobolev spaces $M^{s,p}(...
- 5,405
6
votes
1
answer
284
views
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 ...
- 278
6
votes
1
answer
237
views
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
6
votes
1
answer
257
views
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 (...
- 5,405
6
votes
0
answers
184
views
When is a distance space dominated by a metric space?
A distance space is a pair $(X,d)$ where $X$ is a set and $d:X \times X \rightarrow \mathbb{R}$ is a symmetric, non-negative map such that $d(x,x)=0$ for all $x \in X$. These are sometimes called semi-...
- 193
6
votes
0
answers
182
views
Factorization of metric space-valued maps through vector-valued Sobolev spaces
Let $(X,d,m)$ and $(Y,\rho,n)$ be metric measure spaces and let $f:X\rightarrow Y$ be a Borel-measurable function for which there is some $y_0$ and some $p\geq 0$ such that
$$
\int_{x\in X}\,d(y_0,f(x)...
- 5,405
6
votes
0
answers
813
views
Limit of metric spaces
Let $\{X_n\}_{n\in \mathbb{N}}$ be a collection of T2 topological spaces, with maps $f_n\colon X_n \to X_{n+1}$. These maps are continuous and open. Let $X$ be the direct limit of this system.
Assume ...
- 2,384
5
votes
1
answer
415
views
Spreading $n$ points in $\{0,1\}^n$ as far as possible
Given a positive integer $n$, the Hamming distance $d^H_n(x,y)$ of $x,y\in \{0,1\}^n$ is defined by $$d^H_n(x,y) = |\{k\in\{0,\ldots,n-1\}: x(k)\neq y(k)\}|.$$
We say that a positive integer $s$ is $...
- 48.9k
5
votes
2
answers
2k
views
Isometric embeddings of metric spaces in Hilbert spaces
There are plenty of isometric embeddings of metric spaces in Banach spaces. Nevertheless, I have been unable to find any significant result on isometric embeddings into Hilbert spaces. My question is: ...
- 5,407
5
votes
1
answer
483
views
Can you always extend an isometry of a subset of a Hilbert Space to the whole space?
I remember that I read somewhere that the following theorem is true:
Let $A\subseteq H$ be a subset of a real Hilbert space $H$ and let $f : A \to A$ be a distance-preserving bijection, i.e. a ...
- 609
5
votes
1
answer
200
views
Criterion for Kuratowski Limit Inferior
Let $(X,d_X)$ be a compact metric space and let $\{K_n\}_{n=1}^{\infty}$ be a collection of non-empty compact subsets. Let $K\subseteq X$ be compact. Then, if for every $x_n \in K_n$ we have
$$
d_X(...
5
votes
2
answers
448
views
Space of curves
I am reading Burago, Burago & Ivanov's book where they distinguish the notion of a curve and a path in the following way:
a path in a topological space $X$ is simply a (continuous) map from a ...
- 5,529
5
votes
2
answers
2k
views
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 ...
- 504
5
votes
0
answers
296
views
For which classes of metric spaces can we prove that quasi-isometry is an equivalence relation in ZF?
Given two metric spaces $(M_1, d_1)$ and $(M_2, d_2)$, a map $\phi \colon (M_1, d_1) \to (M_2, d_2)$ is a large-scale Lipschitz essentially surjective map if there exist constants $A \geq 1, B \geq 0$,...
4
votes
2
answers
440
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 ...
- 2,350