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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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22 views

### Do cycle graphs embed isometrically in spheres?

I recently came across, what seems to be a folklore. Namely, that cycle graphs embed, in a bi-Lipschitz manner, with arbitrarily low distortion into spheres of sufficiently high dimension. However, ...
58 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 ...
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### 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 ...
169 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(\...
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### 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 ...
80 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
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### 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 ...
1 vote
37 views

### Metrics on paths in digraphs

I'm looking for metrics (or even just symmetric dissimilarities) on finite paths in finite digraphs but not finding anything in the literature. Can anyone point me to references? I've looked in Deza ...
70 views

### Is there a literature name for this concept of a "graded metric"?

Given a space $X$, I have been thinking about a function $d\colon X \times X \times \mathbb{N} \to \mathbb{R}_{\geq 0}$ (i.e. with values that are nonnegative reals) with the properties below. One may ...
1 vote
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### A generalized Hausdorff dimension in form of a Lower semi continuous function

Let $(X,d)$ be a compact metric space. Assume that $f:X\to \mathbb{R}$ is a positive continuous function. We say that the $f$-dimension of $(X,d)$ is equal to $0$ if for every $\epsilon>0$ ...
1 vote
116 views

### Uniformly open map on a dense subset

Schauder's lemma asserts that you can always extend a uniformly continuous, uniformly open map from a dense subset of a complete metric space to a uniformly open map on the completion. I think the ...
168 views

1 vote
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### Does the topology of Wasserstein space $(\mathcal P_p (E), W_p)$ coincide with the initial topology induced by $\mathcal C_b(E) \cup \{g_p\}$?

Let $(E, d)$ be a Polish space and $\mathcal C_b(E)$ the space of all real-valued bounded continuous functions on $E$. Let $p \in [1, \infty)$ and $\mathcal P_p (E)$ the space of all Borel probability ...
1 vote
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### 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 ...
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### Results in computational geometry utilizing doubling dimension of a metric space

According to Wikipedia, However, many results from classical harmonic analysis and computational geometry extend to the setting of metric spaces with doubling measures. My question is: what are some ...
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### Let $E$ be Banach, $\mu_n\to\mu$ weakly on a locally compact $X$, and $f \in C_b(X, E)$. Does $\int f\mathrm d\mu_n\to\int f\mathrm d\mu$ in norm?

Let $X$ be a metric space, $(E, |\cdot|)$ a Banach space $\mathcal M(X)$ the space of all finite signed Borel measures on $X$, $\mathcal C_b(X)$ be the space of real-valued bounded continuous ...
1 vote
115 views

### Portmanteau theorem for finite signed Borel measures

Let $X$ be a metric space, $\mathcal M(X)$ the space of all finite signed Borel measures on $X$, $\mathcal M_+(X)$ the space of all finite nonnegative Borel measures on $X$, $\mathcal M_1(X)$ the ...
1 vote
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1 vote
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### How to find the point at minimal average distance of a given measure

Given a compactly supported probability measure $m$ on $\mathbb{R}^n$, we can define its average distance to a point $x$ as $\int_\mathbb{R^n}d(x,y)dm(y)$. In this question I found that for a given ...
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 $A\subset Q$ ...
Suppose we are in the following situation: $(X,d)$ is a metric space and $Y$ is a subspace of $X$. Furthermore we have a different metric $\delta$ defined on $Y$ such that $\delta$ is bi Lipschitz ...