# 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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### 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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### 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 ...
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 ...
1 vote
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### 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 $\kappa\in \mathbb{R}$?
1 vote
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### 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 $\epsilon$ ...
1 vote
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### 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 $X$?
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 ...
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 ...
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 $C_d\geq 0$ ...
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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### 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 ...
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&...
1 vote
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### 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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### 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 ...
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 ...
1 vote
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 ...