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 answer
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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\...
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1 vote
1 answer
54 views

Lower bound on a norm of $\mathbb{CP}^2$ inducing a lower bound on the Euclidean norm of $\mathbb{C}^3$

Let $|\cdot|$ denote the usual Euclidean norm on $\mathbb{C}^3$ and fix some arbitrary metric $\rho$ on $\mathbb{CP}^2$. For $\delta > 0$ and any set $\hat{P} \subset \mathbb{CP}^2$, define the $\...
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2 votes
0 answers
40 views

Smallest doubling subset of a set in a metric space

Let $(X,d)$ be a separable metric space and $A\subseteq X$ be compact. Since every finite set is doubling then, the collection $\mathcal{A}$ of doubling subsets of $A$ cannot be empty. My initial ...
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2 votes
1 answer
88 views

A neighborhood $Y$ of a set $X$ such that the line segment connecting any point in $Y$ and its projection to $X$ is contained in $Y$

A direct line from a point $p$ to a set $X$ is a line segment with one endpoint at $p$ and one endpoint in $X$, which is as short as any other line segment from $p$ to $X$. Given a closed set $X$ and ...
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2 votes
0 answers
61 views

Why does the solution to pendulum problem with the geometric approach of Jacobi metric does not correspond to the solution with Lagrangian approach? [closed]

When we solve the pendulum problem with EL equation, we get to the differential equation $\ddot{q}+\frac{g}{l}\sin q=0$ but when I apply the substitution $t \rightarrow t\sqrt\frac{g}{l}$ and ...
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1 vote
0 answers
43 views

Are Carnot groups ever CAT(𝜅) spaces?

Let $G$ be a free Carnot group of homogeneous dimension $d$, equipped with the Carnot–Carathéodory metric. Is $(G,d)$ ever $\operatorname{CAT}(\kappa)$ for some $\kappa\in \mathbb{R}$?
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1 vote
0 answers
145 views

Do all manifolds admit metrics with Euclidean balls?

Let $M$ be a compact topological n-manifold. Suppose we are given a locally flat embedding $M \subset \mathbb{R}^{n+k}$. This induces a metric on $M$ by restriction. Is it true that for $\epsilon$ ...
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1 vote
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89 views

What is the topological characteristic of a separable metric space $X$ such that $|kX\setminus X|=\frak{c}$ for any completion $kX$ of $X$?

What is the topological characteristic of a separable metric space $X$ such that $|kX\setminus X|=\frak{c}$ for any completion $kX$ of $X$?
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9 votes
2 answers
203 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 ...
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2 votes
1 answer
88 views

When are uniform embeddings quasisymetric

Let $X,Y$ be metric space and suppose that $f:X\rightarrow Y$ is a uniform embedding; i.e.: $$ \omega(d_X(x,z))\leq d_Y(f(x),f(z)) \leq \Omega(d_X(x,z)), $$ where $\omega\leq \Omega$ are both strictly ...
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6 votes
1 answer
338 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 $C_d\geq 0$ ...
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3 votes
0 answers
34 views

Estimating the largest radius making each ball in a finite metric space into a tree

Motivation: Let $n$ be a positive integer and $(X,d)$ be an $n$-point metric space. Clearly, $(X,d)$ need not be a metric tree (e.g. take for example the discrete metric on $\{0,1,2\}$. Conversely, ...
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4 votes
0 answers
59 views

Finite approximations to the Kuratowski/Fréchet embedding

Let $(X,d)$ be a compact doubling metric space with doubling constant $C>0$. Let $\{\mathbb{X}_n\}_{n=0}^{\infty}$ be a sequences of finite subsets of $X$ with $$ \left\{B\left(x_k,\frac1{n}\right)...
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4 votes
0 answers
61 views

Covering number $C^k$-balls in $C(\mathbb{R}^n)$

Fix a positive integer $n$ and and an non-negative integer $k$. The Arzela-Ascoli theorem guarantees that for a given positive integer $k$ and a given $L>0$ the set $$ Ball_{C^{k,1}([0,1]^n)}(0,L) ...
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3 votes
4 answers
306 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 ...
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3 votes
0 answers
97 views

Topologically characterizing metrizable spaces

There are some well-known theorems that imply that some metrizable spaces, when satisfying other topological properties, are unique up to homeomorphism. Here are a few examples, where "perfect&...
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1 vote
0 answers
33 views

Stability of Hajłasz-Sobolev class under post-composition

Informally: When is a Sobolev function, post-composed by a vector-valued function still Sobolev? Assumptions/Setup Let $(X,d_X,m_X)$ and $(Y,d_Y,m_Y)$ be complete and separable metric measure spaces; ...
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0 votes
1 answer
77 views

Terminology "upper" Ahlfors regular measure

Let $(X,d)$ be a metric space and $m$ be a Borel measure on $(X,d)$. The measure $m$ is called Ahlors regular if $m(B(x,r))\asymp r^q$ for some $q>0$ and each $x\in X$. Is there a name for ...
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8 votes
1 answer
283 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 ...
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  • 302
1 vote
0 answers
64 views

Best estimate on doubling constant of a finite metric space

Let $(X,d)$ be a finite metric space. Clearly, $(X,d)$ is a doubling metric space but is there a 'best' estimate of $(X,d)$'s doubling constant? Probability based on its cardinality, diameter, and ...
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0 votes
1 answer
167 views

Prove that $(v^Tx)^2-(u^Tx)^2 < 1-(u^Tv)^2$ for any unit vectors $u$, $v$, $x$

Let $u,v,x \in \mathbb R^d$ be three unit vectors. I found a very complicated proof that $(v^Tx)^2-(u^Tx)^2 \leq 1-(u^Tv)^2$. That is $\lVert uu^T-vv^T\rVert^2_2 = 1-(u^Tv)^2$, or that $f(v,x)\leq f(v,...
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11 votes
0 answers
163 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)...
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2 votes
0 answers
113 views

Does a lifted functor on $\mathbf{1Met}$ preserve isometries?

Let $\mathbf{1Met}$ denote the category of metric spaces with distance bounded by $1$ and nonexpansive maps ($1$-Lipschitz functions). I call isometry a distance-preserving map (some people require it ...
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3 votes
1 answer
142 views

Relation between two permutation metrics

Note: I asked this question a few months ago here, but received no answer. Consider the following two metrics on permutations of $\{1,2,\dots,n\}$: $d_\text{swap}(\sigma,\tau)$ is the minimum number ...
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5 votes
0 answers
153 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 &...
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3 votes
0 answers
73 views

String metric properties when extending strings

I am studying some aspects concerning string distance functions, and I am sure there are generic results available in the field of metric spaces, but I have not been able to find appropriate ...
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4 votes
0 answers
134 views

When do Polish spaces admit complete metric making them $\mathrm{CAT}(\kappa)$?

Question $\DeclareMathOperator\CAT{CAT}$Let $X$ be a Polish space. When are there known conditions under which $X$'s topology can be metrized by a metric $d$ such that $(X,d)$ is a: $\CAT(\kappa)$ ...
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1 vote
1 answer
146 views

When are Wasserstein spaces $CAT(\kappa)$?

Let $(X,d)$ be a complete and separable metric space and, for $1\leq p<\infty$, let $(\mathcal{P}_p(X,d),W_p)$ be the $p$-Wasserstein space on $(X,d)$. For which $p$ and $(X,d)$ is $(\mathcal{P}_p(...
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2 votes
0 answers
55 views

Perturbing the approximation property from the Lipschitz-free space to stay in the Wasserstein space

Let $(X,d,x)$ be a separable pointed metric space and let $\mathcal{F}(X)$ be its Arens-Eells (also called its Lipschitz-Free space; in the case where $X$ is Banach) space. We view the $1$-...
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2 votes
0 answers
38 views

A generalization of metrics taking values in partial orders

I'm investigating the origin of the following notion: Let $S=(S, +, <, 0)$ be a partially ordered semigroup with minimum $0$ (such that $<$ is invariant by the action of $+$ on both sides). A $S$...
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  • 383
8 votes
1 answer
326 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 ...
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3 votes
0 answers
64 views

Sufficient conditions for the Besicovitch covering theorem to hold on groups of polynomial growth

Let $G$ be a finitely generated group with symmetric generating set $S$. Then $S$ induces a distance $d$ on $G$ by letting $d(a,b) = $ the minimum $n$ such that there are generators $s_1,...,s_n$ with ...
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4 votes
1 answer
122 views

Axiomatizing projective Hilbert spaces

This question arises in connection to trying to take a different (more intrinsic) perspective on the foundations of quantum mechanics, in which projective Hilbert spaces naturally arise, e.g. see ...
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1 vote
1 answer
62 views

Does the compactness of parameter of distribution function imply the compactness of the distribution (or probability measure) in Wasserstein space?

For a family of probability measures sharing the same form of distribution function $F(x; p)$ with different parameters (i.e., $p$'s), if the parameter falls in a compact subset of real line, can we ...
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5 votes
1 answer
135 views

Compactness of symmetric power of a compact space

Suppose I have a compact metric space $(X,d)$ and let $\mathcal{X}=X^K$ be the product space. Consider the equivalence relation $\sim$ on $\mathcal{X}$ given as: for $\alpha,\beta\in \mathcal{X}$, $\...
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  • 59
1 vote
1 answer
120 views

Effect of snowflaking on doubling constants

This question is related to this one. Let $(X,d)$ be a metric space, let $\epsilon\in [0,1)$ and consider the snowflake $(X,d^{1-\epsilon})$. Suppose that $(X,d)$ has a finite doubling constant, ...
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2 votes
0 answers
87 views

What is the relationship between barycenters in the Arens-Eells sense and barycenters in the optimal transport sense

Setup: Let $X$ be a complete pointed metric space. Let us briefly recall that the Wasserstein space $W_1(X)$ is identifiable with a subset of the Arens-Eells (or Lipschitz-Free) space $\operatorname{...
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1 vote
2 answers
130 views

Monotonicity of doubling dimension

Let $(X,d)$ be a metric space with finite Assouad dimension $0<C_X$. It seems intuitive to me that if $\emptyset \subset Y\subseteq X$ then $Y$ is also doubling and its Assouad dimension, denoted ...
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2 votes
3 answers
234 views

Banach-like analysis on metric spaces

Some time ago, I was thinking about whether it would be possible to generalize some results from functional analysis on Banach spaces to some metric spaces. Specifically, I wondered whether if one ...
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2 votes
0 answers
81 views

Almost Lipschitz embedding of compact metric measure spaces into Euclidean spaces

Let $(X,d)$ be a compact metric space, $m$ be a metric outer-measure on $X$. Are there 'mild conditions' on $X$ ensuring the existence of a positive integer $N\geq 3$ such that there exist $x_1,\dots,...
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8 votes
0 answers
192 views

Covering number estimates for Hölder balls

Let $\alpha \in (0,1]$ and $L>0$. The Arzela-Ascoli Theorem guarantees that the set $X(\alpha,L)$ of $f:[-1,1]^n\rightarrow \mathbb{R}^m$ with $\alpha$-Hölder norm at-most $L$ is compact in $C([-1,...
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0 votes
0 answers
45 views

Holder-continuous barycenter maps

Let $(X,d)$ be a complete locally-compact metric space. We define the $p$-barycenter map as a continuous function: $$ \beta:\mathcal{P}_p(X)\rightarrow X, $$ which is a right-inverse of the map ...
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0 votes
0 answers
71 views

Relationship between various notions of laws of random functions

Let $X,Y,Z$ be locally-compact metric spaces, $f:X\times Y\rightarrow Z$ be uniformly continuous, and let $\xi$ be an $X$-valued random element. Equip the set $C(Y,Z)$ with cylindrical $\sigma$-...
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2 votes
0 answers
107 views

Relationship between Hausdorff dimension and covering number

Let $(X,d)$ be a compact metric space and recall that the $\epsilon$-external covering number $\mathcal{N}^{\epsilon}(X)$ of $X$ is defined by: $$ \mathcal{N}^{\epsilon}(X) := \inf\left\{ N\in \mathbb{...
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1 vote
0 answers
56 views

Injectivity of post-composition operator

Let $X$, $Y_1,Y_2$, and $Z$ be separable metric spaces. Let $C(X,Y)$ be the topological space of continuous functions from $X$ to $Y$ equipped with its compact-open topologies. Fix a continuous ...
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9 votes
1 answer
252 views

Continuously varying norms

Let $V$ be a finite-dimensional real vector space with its Euclidean topology. Then all norms on $V$ are equivalent and consequently given two norms $\lVert-\rVert$, $\lVert-\rVert'$, the number $$ d =...
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6 votes
1 answer
255 views

How complicated can the path component of a compact metric space be?

Let $X$ be a compact metric space and $P$ be a path component of $X$. Since we are not assuming $X$ is locally path connected, $P$ must need not be open nor closed. Certainly, $P$ must be separable ...
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0 votes
0 answers
85 views

Banach fixed point theorem / convergence squeeze

I am trying to prove a convergence result on an iterative scheme which has the initial point defined as $$x_1 = \frac{1 - s(x_0)}{s(x_0)}$$ where s(x) is some unknown function. Here is my theorem and ...
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3 votes
0 answers
126 views

Are there any major differences in metric topologies and "non-symmetric" metric topologies

Let $X$ be a set and let $d:X\times X\rightarrow [0,\infty)$ satisfy all the axioms of a metric besides symmetry (i.e.: $d$ is a quasi-metric). Define a topology $\tau_{d:+}$ on $X$ induced by $d$ as ...
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4 votes
0 answers
91 views

"Snowflaked" Hausdorff metric

Let $(X,d_X)$ be a compact metric space and let $Comp(X)$ be the set of closed subsets of $X$ with the Hausdorff metric: $$ D(A,B)\overset{\text{def}}{=} \, \max\left\{\sup_{b\in B}\,d_{A}(b),\sup_{a\...
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