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
2
votes
2
answers
297
views
Density of subsequences in Bolzano-Weierstrass
Let $(M, d)$ be a metric space and $K$ compact. It is known that $K$ is sequentially compact, so we can "run" Bolzano-Weierstrass on it.
I want to identify the set $\mathcal{F}$ of all ...
- 83
13
votes
2
answers
765
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 ...
- 43.2k
7
votes
0
answers
150
views
The space of analytic associative operations
This question is a follow-up to this old one of mine.
Let $\mathcal{A}$ be the set of functions $\star:\mathbb{R}^2\rightarrow\mathbb{R}$ which are associative and $C^\omega$ (real analytic entire) in ...
- 20.9k
4
votes
0
answers
197
views
Compactness of the unit ball in the space of Radon measures w.r.t. the Kantorovich-Rubinstein norm
This question was posted previously but has not attracted any responses so I am repharising it in a slightly different language hoping to reach a wider community
Let $(X,d)$ be a pointed metric space ...
- 437
2
votes
2
answers
226
views
A property for maps between metric spaces
Let $X, Y$ be metric spaces with distance functions denoted by $d_X, d_Y$ respectively. Consider a map $f \colon X \rightarrow Y$. I am interested in the following property: for every $x,y,z \in X$, ...
- 327
1
vote
0
answers
178
views
Trans-universality for finite-dimensional Banach space
In addition to a specific problem Trans-universality for finitely generated groups, I posted also its general form. It should not hurt to provide another special case:
QUESTION: does there exist a ...
- 4,786
0
votes
1
answer
238
views
Trans-universality for finitely generated groups
QUESTION: does there exist a group U such that three conditions hold:
(a) every finitely generated group is isomorphic to a subgroup of U;
(b) for every group G that is not finitely generated there ...
- 4,786
1
vote
1
answer
116
views
Do Gromov hyperbolic spaces admit concical geodesic bicombings?
Consider a metric space $(X,d)$ with a distinguished selection of geodesics, i.e. a geodesic bicombing $\sigma:X\times X\times [0,1]\rightarrow X$. We call a geodesic bicombing conical if it ...
- 364
1
vote
1
answer
44
views
Graphs admitting an 1-Lipschitz map from edge mid-points to vertices
Let $G=(V,E)$ be a graph. A 1-Lipschitz vertex projection is a map $p: E \to V$ such that $p(e)$ is always an end-vertex of $e$, and if $e,f$ have a common end-vertex, then $p(e)$ and $p(f)$ coincide ...
- 1,926
0
votes
1
answer
410
views
Properties of doubling metric spaces
At present I work with tools that involves doubling metric space, my definition of DME is:
A metric space $X$ is called doubling with constant $N$, where $N \geq 1$ is an integer, if, for each ball $...
- 101
1
vote
0
answers
30
views
Can we modify this extended pseudometric such that its convergence is equivalent to that in measure?
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ be a complete $\sigma$-finite measure space,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of $\mu$-simple ...
- 657
2
votes
0
answers
292
views
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 ...
- 1,341
1
vote
0
answers
87
views
Is this metric on the space of $\mu$-measurable functions complete?
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ be a complete finite measure space,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of $\mu$-simple functions ...
- 657
4
votes
1
answer
210
views
Bi-Lipschitz embeddings of compact doubling spaces
Suppose that $(X,\rho)$ is a compact doubling metric space. Does there necessarily exist an $\epsilon>0$ and a maximal $\epsilon$-net $\{x_i\}_{i=1}^n\subseteq X$ such that the map
$$
\begin{...
- 5,405
2
votes
1
answer
46
views
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 ...
1
vote
1
answer
170
views
Billingsley convergence of probability measures - inequality used in Theorem 2
On Page 8, Billingsley defines $f(x)=(1-\rho(x,F)/\epsilon)^{+}$ where $\rho(x,F)$ is the metric distance from the set $F$. He then states $|f(x)-f(y)|\leq \rho(x,y)/\epsilon$ and goes on to use this ...
- 19
1
vote
0
answers
126
views
Absolute continuity of the volume growth in a metric space
Let $(M,d)$ be a metric space (separable, complete, better?) and let $\mu$ be a ($\sigma$-additive, positive, locally finite, regular?) Borel measure on $M$. For $x\in M$ consider the volume growth ...
- 1,959
1
vote
1
answer
97
views
Expected value of the projective metric between random orthogonal Stiefel matrices in $\mathbb{R}^{N \times k}$ equals $1 - \frac{k}{N}$
This is a cross-post from this other question that I asked ~1 month ago in the mathematics forum, with no reaction. I am still stuck on this, looking for references or approaches to proofs. I hope I ...
- 113
3
votes
0
answers
61
views
Isometric embedding of 4-element metric spaces into Riemannian manifolds and the curvature
I came across this question Preferred embedding of finite metric spaces in riemaniann manifolds of given dimension. In one of the answers it was stated that it is always possible to isometrically ...
- 820
0
votes
1
answer
131
views
Is this a smooth approximation to the $\ell$-infinity distance actually a quasi-metric?
The $\|\cdot\|_{\infty}$-norm on $\mathbb{R}^n$ for $n\in \mathbb{Z}^+$ is not a smooth function. However, I came across this post which essentially says that a pointwise approximation to the maximum ...
- 169
2
votes
0
answers
48
views
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 ...
- 429
0
votes
1
answer
115
views
Generalized Triangle Inequality for Snowflakes
Let $p>0$ and consider a metric space $(X,d)$. I have recently come across a problem where the space $(X,d^q)$ provides is natural; where $q>1$. However, the triangle inquality break (i.e. it ...
- 169
2
votes
0
answers
47
views
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\...
1
vote
0
answers
125
views
Do cycle graphs embed isometrically in spheres?
I recently came across, what seems to be a folklore. Namely, that cycle graphs embeds isometrically into spheres $S^n(r)$, for some $n\in \mathbb{N}_+$ and some $r>0$. However, I could not track ...
- 169
3
votes
0
answers
69
views
Is every weakly $1$-dimensional space embeddable in the plane?
A $1$-dimensional (separable metric) space $X$ is weakly $1$-dimensional if $$\Lambda(X)=\{x\in X:X\text{ is 1-dimensional at }x\}$$
is zero-dimensional (i.e. the space $\Lambda(X)$ has a basis of ...
- 3,317
6
votes
0
answers
155
views
Metric spaces containing a topological disc
It is well-known that every connected, locally connected compact metrizable space $X$ contains an arc, that is, a subspace homeomorphic to $[0,1]$. Are there topological properties we can add to these ...
- 7,193
5
votes
1
answer
198
views
Iterating the dimensional kernel of a metric space
Fix $n\in \mathbb N$. Let $X$ be a separable metric space of (inductive) dimension $n$. Let
\begin{align}
\Lambda(X)&=\{x\in X:X\text{ is $n$-dimensional at }x\}\\ \\
\Lambda^2(X)&=\Lambda(\...
- 3,317
1
vote
1
answer
276
views
Defining area / n-volume of a finite metric space
Let $(X, d)$ be a finite metric space. I've seen several answers to the question when can $X$ be isometrically embedded into Euclidean space (or, more generally, Riemannian manifold). I'm interested ...
- 820
4
votes
1
answer
104
views
Generalization of a bounded variation
Let $(X, d)$ be a metric space. We will say that $\gamma \colon [a,b] \to X$ is of bounded variation, if
\begin{equation}
V(\gamma) = \sup_{a=t_0 < \cdots < t_n < b} \sum_{i=1}^n d( \gamma(...
- 820
1
vote
0
answers
62
views
What's the best arrangement of solid objects for providing shade?
Let's say we have a horizontal roof and the sun is going to go from 0 to some number of degrees on the horizon. We wish to arrange solid objects above the roof to completely block out the sun across ...
- 31
1
vote
0
answers
48
views
Metrics on paths in digraphs
I'm looking for metrics (or even just symmetric dissimilarities) on finite paths in finite digraphs but not finding anything in the literature. Can anyone point me to references?
I've looked in Deza ...
- 15.4k
2
votes
0
answers
74
views
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 ...
- 21
1
vote
1
answer
112
views
Changing a metric to that 2 points have different distance
Let $X$ be a compact metric space. Assume that $X$ has more than $2$ points (or even better, that $X$ is connected with more than 1 point). Given a metric $d$ on $X$ we define $$d(x,X)=\max\{d(x,z):z\...
- 3,317
2
votes
0
answers
130
views
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 ...
- 820
3
votes
1
answer
132
views
If $X,X'$ have the same $\varepsilon$-packing numbers and $f:X \to X'$ surjective $1$-Lipschitz, then $f$ is an isometry
Let $(X, d)$ be a compact metric space.
We say that $\{x_1, \cdots, x_n\} \subseteq X$ is an $\varepsilon$-covering of $X$ if for any $x \in X$, there exists $i \in \{1, \ldots, n\}$ such that $d(x, ...
- 835
2
votes
1
answer
139
views
Are two metric spaces isometric if they have the same $\varepsilon$-covering and $\varepsilon$-packing numbers for all $\varepsilon>0$?
Let $(X, d)$ be a compact metric space.
We say that $\{x_1, \cdots, x_n\} \subseteq X$ is an $\varepsilon$-covering of $X$ if for any $x \in X$, there exists $i \in \{1, \ldots, n\}$ such that $d(x, ...
- 835
2
votes
1
answer
259
views
Are two metric spaces isometric if they have the same $\varepsilon$-covering numbers for all $\varepsilon>0$?
Let $(E, d)$ be a metric space. For $\varepsilon>0$, we define two notions of $\varepsilon$-covering number as follows, i.e.,
$N_\varepsilon^o (E)$ is the smallest number of open balls whose radii ...
- 835
4
votes
1
answer
292
views
Is every 1-Lipschitz homeomorphism $f:X\to X$ from a compact metric space to itself an isometry?
I found a statement involving a homeomorphism $f:X\to X$ of a compact metric space $X$, with Lipshitz coefficient 1, i.e., a non-expansive map, and cannot think of an example where $f$ is not an ...
- 10.6k
3
votes
1
answer
150
views
Is this map (from the space of probability densities to the Wasserstein space) Lipschitz?
Let $p \in [1, \infty)$. Let $\mathcal P_p(\mathbb R^d)$ be the space of all Borel probability measures on $\mathbb R^d$ with finite $p$-th moments. Let $D_p$ be the collection of all Borel measurable ...
- 657
15
votes
1
answer
684
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 ...
- 708
0
votes
1
answer
91
views
Can a measure on a finite metric space be Alhfors regular?
Recall that a probability $\mu$ measure on a metric space $(X,d)$ is called Ahlfors $q$-regular if there are $0<c\le C$ such that: for $\mu$-a.e.\ $x\in X$ one has
$$
cr^q \le \mu(B(x,r)) \le Cr^q,
...
- 364
6
votes
0
answers
111
views
A generalized Hausdorff dimension in form of a Lower semi continuous function
Let $(X,d)$ be a compact metric space. Assume that $f:X\to \mathbb{R}$ is a positive continuous function. We say that the $f$-dimension of $(X,d)$ is equal to $0$ if for every $\epsilon>0$ ...
- 356
1
vote
0
answers
165
views
Uniformly open map on a dense subset
Schauder's lemma asserts that you can always extend a uniformly continuous, uniformly open map from a dense subset of a complete metric space to a uniformly open map on the completion.
I think the ...
- 11
2
votes
0
answers
238
views
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 ...
- 1,451
2
votes
1
answer
110
views
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?
- 5,405
0
votes
0
answers
131
views
Cyclic group action and finite invariant set
Let $(X, d)$ be a compact metric space and $G$ a discrete group acting on $X$ such that, for each $g\in G$, the mapping $x\mapsto g\cdot x$ defines a homeomorphism on $X$
Is it true that the ...
- 307
1
vote
1
answer
164
views
Right-continuity of covering number
Consider an ambient metric space $(\mathcal{X},\Vert\cdot\Vert_\infty)$. Let $\mathcal{B}_1 = \mathcal{B}_{\Vert\cdot\Vert_K}(0,1)\subseteq\mathcal{X}$ be the closed unit ball with respect to some ...
- 23
1
vote
1
answer
284
views
Bilipschitz embedding of the unit ball of $c_0$ into $\ell_1$
This is a follow-up to this question of mine.
It is well-known that the Banach space $\ell_1$ does not contain any isomorphic copies of $c_0$. One can even go further and show that $\ell_1$ does not ...
- 1,151
1
vote
1
answer
221
views
What properties are preserved by quasi-isometries
Recently, I came across the notion of quasi-isometries, while thinking of "discrete spaces which are surrogates for approximate continuous ones".
What (metric)/geometric properties are ...
- 5,405
2
votes
0
answers
150
views
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 \...
- 657