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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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Partition of unity without AC

Several existence theorems for partition of unity are known. For example (source), Proposition 3.1. If $(X,\tau)$ is a paracompact topological space, then for every open cover $\{U_i \subset X\}_{i \...
BonBon's user avatar
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2 votes
1 answer
144 views

A topological characterization of trees?

Motivated by this complex dynamics question: Let $X$ be a compact, path-connected metric space. Suppose there exist an integer $N\geq 2$ and distinct points $p_1,\dots,p_N\in X$ such that no proper ...
KhashF's user avatar
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1 vote
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Genaralizing the metric expression present in the quadrilateral inequality

Let $(X, d)$ be a metric space. In Sato - An alternative proof of Berg and Nikolaev’s characterization of CAT(0)-spaces via quadrilateral inequality it is stated that if $X$ is a geodesic space, then ...
Kacper Kurowski's user avatar
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24 views

Definition of semi-metric for empirical process theory

In the following lecture notes on empirical processes (https://www.stat.columbia.edu/~bodhi/Talks/Emp-Proc-Lecture-Notes.pdf) a semi-metric space $(\Theta, d)$ is defined in the following way: for any ...
Stan's user avatar
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3 votes
1 answer
179 views

Does complete and separable Wasserstein space imply a complete base space?

Also asked on math.SE. Let $(Z,d)$ be a metric space, and for $p\geq 1$, consider a metric space $(W_p,d_{W_p})$ defined by The Wasserstein Space $\begin{align}W_p = \{\mu|\mu\textrm{ is a Borel ...
Kaira's user avatar
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0 answers
44 views

Any useful bases for the topology induced by the $t$-Wasserstein distance?

I am working on $\mathbb R ^d$ equipped with the usual Euclidean metric. I know of one nice base for $\mathcal W _t$, namely: $$\left\{ B_p (r) : r>0, p=\sum_{i=1} ^n \alpha_i \delta_{x_i},\text{ ...
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0 answers
54 views

Wasserstein space isomorphic to original space?

Is there a complete measurable metric space $(X,d)$ for which its $p$-Wasserstein space $W(X)$ is isometrically isomorphic to $(X,d)$ for some $p \in [1,\infty]$? Note that there is a canonical non-...
Florentin Münch's user avatar
2 votes
1 answer
71 views

Hausdorff-Lipschitz continuity of cone correspondence

Let $\mathbb{R}_+$ denote the strictly positive real numbers, let $\mathcal{X} \subset \mathbb{R}^n$ and $\mathcal{P} \subset \mathbb{R}^m$ be compact and convex subsets, let \begin{equation} f: \...
Heinrich A's user avatar
2 votes
0 answers
82 views

For Polish $X,Y$, $L^p(X,Y)$ is separable

Let $X$ and $Y$ be Polish spaces. Equip $X$ with a Borel probability measure $\mu_X$ and $Y$ with a metric $d_Y$. We can define the $L^p$ space as follows: Definition. Define $\begin{align}L^p(X,Y) = \...
Kaira's user avatar
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-4 votes
1 answer
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Can a Polish space have two different topologies?

Let $X$ be a Polish space with the compatible metric being $d_1$. So $(X,d_1)$ is a separable complete metric space, and the topology is generated by $d_1$. Can there be a metric $d_2$ such that $(X,...
J.R.'s user avatar
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3 votes
1 answer
177 views

More than one recurrence point (Birkhoff)

Birkhoff's recurrence theorem states that for a compact metric space $X$ and a continuous function $T: X\rightarrow X$, there is a recurrence point $x\in X$; the latter means that for any ...
Sam Sanders's user avatar
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1 vote
1 answer
67 views

Lipschitz approximation of a probability measure with finite $1$-st moment by the ones with finite $p$-th moment

For $p \in [1, \infty)$, let $\mathcal P_p (\mathbb{R^d})$ be the space of Borel probability measures on $\mathbb R^d$ with finite $p$-th moment. We endow $\mathcal P_p (\mathbb{R^d})$ with the ...
Akira's user avatar
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1 answer
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Variants of Dirichlet-type function as a pointwise limit of continuous functions

Problem Suppose $f$ is a function from a complete metric space $X$ to a metric space $Y$, and suppose $Y$ has points $y_{0}$, $y_{1}$ such that the subsets $f^{-1}(y_{0})$ and $f^{-1}(y_{1})$ are both ...
hmeng's user avatar
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3 votes
0 answers
109 views

Is the Schwartz space a tame Frechet space?

I ran into the following definition of tame Frechet spaces and Nash-Moser therem. It says that the space of smooth functions on a compact manifold is tame Frechet. However, I wonder if The Schwartz ...
Isaac's user avatar
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-2 votes
1 answer
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Interpretation and validity of modified Heisenberg uncertainty principle in a metric context? [closed]

Considering the Heisenberg uncertainty principle, which states $\Delta x \cdot \Delta p \geq h$, I've explored a modified version by computing $(\Delta x + 1)(\Delta p + 1) \geq \Delta x \cdot \Delta ...
mathoverflowUser's user avatar
6 votes
0 answers
181 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-...
David Bryant's user avatar
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0 answers
310 views

The set of continuous bounded functions $f:X\to Y$ is dense in $L^p(X,Y)$ where $X,Y$ are Polish

It is well known that the set of real-valued continuous functions with compact support is dense in $L^p(\mu)$ where $\mu$ is a Radon measure (see e.g. [Folland, Proposition 7.9]) Clearly, the set of ...
Kaira's user avatar
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2 votes
0 answers
184 views

Statistical invariants of Riemannian manifolds

$\DeclareMathOperator\diam{diam}\DeclareMathOperator\rad{rad}\DeclareMathOperator\iso{iso}\DeclareMathOperator\com{com}\DeclareMathOperator\con{con}$A cheap way of defining invariants of Riemannian ...
Alex's user avatar
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2 votes
0 answers
83 views

Can we control the Wasserstein metric between $\mu$ and $\nu$ by their moment difference?

Fix $p \in [1, \infty)$. Let $(\mathcal P_p(\mathbb R^d), W_p)$ be the Wasserstein space of all Borel probability measures on $\mathbb R^d$ with finite $p$-th moment. Let $D_p$ be the collection of ...
Analyst's user avatar
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0 answers
90 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 ...
JE2912's user avatar
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3 votes
1 answer
153 views

Equivalent definition for Skorokhod metric

I have a question about the Skorokod distance on the space $\mathcal{D}([0,1],\mathbb{R})$: $$ d(X,Y):= \inf_{\lambda \in \Lambda}\left( \sup_{t\in [0,1]}|t-\lambda(t)|\vee \sup_{t\in [0,1]}|X(t)-Y(\...
user1598's user avatar
  • 177
31 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,...
aleph2's user avatar
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0 votes
1 answer
87 views

Kähler metric on the projective space

"Is there a Kähler metric on the complex projective space $\mathbb {P} ^n(\mathbb {C} ) $ different from the Fubini-Study metric?
Samir's user avatar
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0 answers
67 views

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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1 vote
1 answer
138 views

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
3 votes
0 answers
177 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
2 votes
2 answers
285 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 ...
Daniel Goc's user avatar
13 votes
2 answers
728 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
7 votes
0 answers
147 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
0 answers
182 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
0 votes
0 answers
34 views

Existence of a minimal, weakly mixing and Lipschitz selfmap?

I am looking for an example of a dynamical system $(M,f)$ such that: $M$ is a metric space; $f:M \to M$ is Lipschitz; $f$ is weakly mixing (that is $f \times f$ is topologically transitive) $f$ is ...
Baguette's user avatar
2 votes
2 answers
223 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 $x,y,z \in X$, ...
gm01's user avatar
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1 vote
0 answers
177 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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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 ...
Wlod AA's user avatar
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1 vote
1 answer
108 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
40 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
1 answer
232 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 $...
C L 's user avatar
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1 vote
0 answers
29 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 ...
Analyst's user avatar
  • 647
2 votes
0 answers
291 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
0 answers
81 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 ...
Analyst's user avatar
  • 647
3 votes
1 answer
180 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
  • 4,809
2 votes
1 answer
44 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
1 vote
1 answer
160 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 ...
Simon's user avatar
  • 19
1 vote
0 answers
120 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
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1 vote
1 answer
87 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
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3 votes
0 answers
59 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
0 votes
1 answer
102 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
2 votes
0 answers
46 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
  • 419
0 votes
1 answer
99 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
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\...
pyridoxal_trigeminus's user avatar

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