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

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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 $x,y,z \in X$, ...
gm01's user avatar
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1 vote
1 answer
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Divergence functions in hyperbolic groups

Gromov hyperbolicity has many characterizations, one of them being the existence of a super-linear divergence function, see definition below. We note that in $\mathbb{R}^2$ there is no divergence ...
Strichcoder's user avatar
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 ...
Julian Newman's user avatar
3 votes
0 answers
239 views

Metrizing pointwise convergence of *sequences* of functionals in a dual space

This question was asked by myself on the math stackexchange a few days ago. I thought I'd repeat it here: Let $X$ be a normed, real vector space of uncountable dimension. Let $X^*$ denote the set of ...
Mustafa Motiwala's user avatar
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 ...
Noah Schweber's user avatar
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{...
ABIM's user avatar
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0 votes
0 answers
117 views

Example of a metrizable space that is not an ANR

I have been looking for an example of a metrizable space that is not an absolute neighborhood retract (ANR). Recall that a metrizable space $X$ is called an ANR if there exists an open set $U$ in a ...
Katrina's user avatar
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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 ...
Yury Korolev's user avatar
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 ...
Wlod AA's user avatar
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2 votes
2 answers
293 views

Optimal transport: the existence of an optimal pair of $c$-conjugate functions

$\newcommand{\diff}{ \, \mathrm d}$ Let $X,Y$ be Polish spaces, $\mathcal C_b(X)$ the space of all real-valued bounded continuous functions on $X$, $\mathcal P(X)$ the space of Borel probability ...
Akira's user avatar
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0 answers
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What is a metric for weak convergence of finite measures on a non compact, complete and separable metric space?

Consider the set of finite positive measures on a complete, separable, but not compact, metric space $S$, endowed with the topology under which a sequence of finite positive measures $\{\mu_n\}$ ...
CCC's user avatar
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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 ...
Denis's user avatar
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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 ...
Math_Newbie's user avatar
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 ...
Kacper Kurowski's user avatar
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, ...
Akira'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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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(\...
D.S. Lipham's user avatar
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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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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 ...
Justin_other_PhD's user avatar
9 votes
4 answers
4k 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
  • 657
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 ...
pyridoxal_trigeminus's user avatar
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 ...
Akira's user avatar
  • 835
6 votes
2 answers
424 views

Lipschitz mappings, covering dimension

Is there a compact metric space $X$ of covering dimension $2$ without a Lipschitz surjection on $[0,1]^2$? For a space $X$ with Hausdorff dimension greater than $2$, we have a negative answer (see ...
Hpela's user avatar
  • 97
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 ...
Saúl RM's user avatar
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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\...
M. Winter's user avatar
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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 ...
Agelos's user avatar
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0 votes
0 answers
65 views

Random covering on rectangles

Let $\mathrm{Rect}$ denote the class of axis-parallel rectangles $r: \mathbb{R}^2 \to \{0,1\}$, assigning $1$ if the point is inside the rectangle and $0$ otherwise. Let $\mathcal{D}$ be a ...
Saginus's user avatar
  • 159
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 ...
fr_andres's user avatar
  • 113
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 ...
Damian Sobota's user avatar
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 ...
Bedovlat's user avatar
  • 1,959
3 votes
4 answers
367 views

Prove that $(v^Tx)^2−(u^Tx)^2\leq \sqrt{1−(u^Tv)^2}$ for any unit vectors $u, v, x$

I believe I found a complicated proof by bounding the spectral norm $||uu^T-vv^T||^2_2:=\max_{||x||=1}|(u^Tx)^2-(v^Tx)^2|$. Using the fact that $dist(x,y):=\sin|x-y|$ is a distance function over unit ...
Dan Feldman's user avatar
4 votes
2 answers
191 views

Reference request: "Tangent relation" in metric spaces

Let $X,Y$ be metric spaces. Let $f,g : X \to Y$ be two maps and $x_0 \in X$. Let us say that $f$ and $g$ are tangent at $x_0$ if the following condition is satisfied: For every $\epsilon > 0$ there ...
Martin Brandenburg's user avatar
35 votes
6 answers
2k views

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 ...
user avatar
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 ...
Justin_other_PhD's user avatar
2 votes
1 answer
336 views

Complex Borel measures: does $\mu_n \to \mu$ weakly imply $|\mu|(\Theta) \le \liminf_n |\mu_n|(\Theta)$ for every open subset $\Theta$?

Let $\Omega$ be a metric space, $C_b(\Omega)$ the space of all real-valued bounded continuous functions on $\Omega$, and $\mathcal{M}(\Omega)$ the space of all finite signed Borel measures on $\Omega$...
Analyst's user avatar
  • 657
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 ...
Justin_other_PhD's user avatar
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 ...
Kacper Kurowski's user avatar
3 votes
1 answer
486 views

There exists differentiable curves arbitrarily close to the continuous ones

Let $M$ be a Riemannian manifold; if $d$ is the distance on $M$, we can consider the distance $D$ between any two continuous curves given by $D(c, \gamma) = \max _{t \in [0,1]} d(c(t), \gamma(t))$. ...
Alex M.'s user avatar
  • 5,407
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(...
Kacper Kurowski's user avatar
4 votes
2 answers
374 views

Vague convergence: confusion about the regularity of a signed Radon measure and that of its variation

I'm reading a proof of below theorem from this paper. Theorem A.3. Let $\Omega$ be a locally compact normal Hausdorff space. Let $\left\{\mu_n\right\} \cup\{\mu\} \subset \mathcal{M}(\Omega)$ and ...
Analyst's user avatar
  • 657
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 ...
Jeremy Brazas's user avatar
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 ...
Analyst's user avatar
  • 657
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, ...
Akira's user avatar
  • 835
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 ...
omar's user avatar
  • 278
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 ...
Marco Ripà's user avatar
  • 1,451
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 $A\subset Q$ ...
Anton Petrunin's user avatar
7 votes
3 answers
340 views

Hyperbolic space embeds into Wasserstein space

Fix a positive integer $n$, let $\mathbb{H}^n$ be the $n$-dimensional hyperbolic space, $r>0$, $x\in \mathbb{H}^n$ and consider the closed (compact) geodesic ball $B_{\mathbb{H}^n}(x,r)$. Are ...
Carlos_Petterson's user avatar
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 ...
D.S. Lipham's user avatar
  • 3,317
8 votes
3 answers
937 views

BCT equivalent to DC

Do you know where I can find proof of equivalence Baire Category Theorem and DC (Axiom of Dependent Choice)? It is well known fact but I can't find appropriate literature with the proof.
Michael's user avatar
  • 143
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 ...
alhal's user avatar
  • 429

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