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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1 vote
1 answer
99 views

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 ...
2 votes
0 answers
90 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 ...
-2 votes
1 answer
122 views

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 ...
6 votes
0 answers
176 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-...
0 votes
0 answers
165 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 ...
1 vote
2 answers
475 views

Is there good evidence that topological spaces are the correct way to study the general theory of continuity? [closed]

My reason for asking is that the theory of metric spaces is so clean and so many significant theorems can be proved for an arbitrary metric space (which makes it plausible to me that metric spaces are ...
2 votes
0 answers
176 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 ...
2 votes
0 answers
73 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 ...
0 votes
1 answer
142 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 ...
3 votes
1 answer
147 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(\...
0 votes
0 answers
75 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 ...
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,...
5 votes
1 answer
331 views

Points of differentiability of squared distance from a point in metric spaces

I posted this same question on MSE with no answer. Let $I:=(0, + \infty)$ and let $(X,d)$ be a complete and separable metric space. In this setting we say that $u : I \to X$ is absolutely continuous ...
0 votes
1 answer
76 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?
0 votes
0 answers
58 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 $\{\mu_n\}$ ...
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 ...
1 vote
1 answer
116 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 ...
3 votes
0 answers
151 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 ...
2 votes
2 answers
274 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 ...
4 votes
0 answers
165 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 ...
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 ...
7 votes
0 answers
146 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 ...
0 votes
1 answer
272 views

Does there always exists a path $g:[0,1] \rightarrow X$ from $f(0)$ to $f(1)$ that has the same image as $f$ and …?

Suppose $(X,d)$ is a metric space and $f:[0,1] \rightarrow X$ is a path in $X$ with non-zero finite length $L$. Then, does there always exist a path $g:[0,1] \rightarrow X$ from $f(0)$ to $f(1)$ that ...
0 votes
0 answers
33 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 ...
2 votes
2 answers
220 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 $x,y,z \in X$, ...
1 vote
0 answers
174 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 ...
0 votes
1 answer
237 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 ...
1 vote
1 answer
100 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 ...
1 vote
1 answer
212 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 ...
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 ...
1 vote
1 answer
39 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 ...
0 votes
0 answers
129 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 $...
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 ...
2 votes
0 answers
290 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 vote
0 answers
77 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 ...
4 votes
1 answer
165 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{...
1 vote
1 answer
75 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 ...
1 vote
0 answers
118 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 ...
3 votes
0 answers
56 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 ...
0 votes
1 answer
87 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 ...
2 votes
0 answers
44 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 ...
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(\...
13 votes
0 answers
634 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 $\alpha$-...
0 votes
1 answer
95 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 ...
1 vote
0 answers
118 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 ...
2 votes
0 answers
46 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\...
3 votes
0 answers
68 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 ...
6 votes
0 answers
144 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 ...
4 votes
1 answer
94 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(...
1 vote
0 answers
61 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 ...

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