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) $d(x,y)+d(y,z) \geq d(x,z)$.
442 questions
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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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Earth movers distance (EMD) between two multivariate normals. Is it negative definite distance?
I was looking at the closed form formula for 2-Wassersteins distance for multivariate normal distribution on Wikipedia. https://en.wikipedia.org/wiki/Wasserstein_metric#Normal_distributions
It has a ...
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Length spaces with continuous length functional: is this set Gromov-Hausdorff closed?
As far as I can tell, a major motivation for the study of length spaces is that they arise as Gromov-Hausdorff limits of Riemannian manifolds. Specifically,
A complete connected Riemannian manifold ...
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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 ...
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cardinality of local bases in the non-standard reals
Given a index set $S$ together with a ultrafilter $\mu$ on $S$ (such that no set of cardinality $< |S|$ has measure $1$). Let the ordered field $\mathbb{R}(S,\mu)$ denote the ultrapower of $\mathbb{...
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radius-diameter relationship of balls in metric spaces
What necessary and sufficient conditions must a metric $d$ of a metric space $(X,d)$
fulfill so that the open balls of radius $r$ have diameter $2r$?
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(non) separability of the power set
Suppose that $(X,2^X)$ is equipped with a non-atomic probability measure $\mu$ (the existence of such spaces is consistent with ZFC). This induces the $L_1$ pseudometric $\Delta$ on $2^X$, via
$\...
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Does uniform continuity of bounded continuous functions implies the same for all continuous functions on a uniform space?
Recently I came to know about Atsuji space from the paper1. A metric space $X$ is called an Atsuji space if every real-valued continuous function on $X$ is uniformly continuous. Strikingly I have ...
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Takahashi convex metric spaces
A Takahashi convex metric space is a metric space $(X,d)$ such that $\exists W : X \times X \times [0,1] \rightarrow X$ that satisfies :
$d (u, W(x,y; \lambda)) \leq \lambda d(u,x) + (1- \lambda) d(u,...
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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 ...
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Disintegration of measures: a confusion about an existence proof from a lecture note
I'm reading a proof of Theorem 2.25 below from this note. First, we recall a definition and a theorem, i.e.,
Theorem 2.25 (Disintegration). Let $\left(Z, d_Z\right)$ and $\left(X, d_X\right)$ be ...
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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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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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"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:
$$
D(A,B)\overset{\text{def}}{=} \, \max\left\{\sup_{b\in B}\,d_{A}(b),\sup_{a\...
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Relationship between Hausdorff convergence of sets and indicator functions
Let $\{K_n\}_n$ be a sequence of compact subsets of a metric space $X$, and $K\subset X$ be compact. If $K_n$ Hausdorff converges to $K$, i.e.:
$$
\lim\limits_{n\to\infty} d_{\mathrm H}(K_n,K) = \max\...
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Continuous extension preserving modulus of continuity
Let $X$ be a (non-empty) compact subset of $D(0,M):=\left\{x\in \mathbb{R}^n:\, \|x\|\leq M\right\}$, and let $f:X\rightarrow Y$ be uniformly continuous; for some metric space $Y$. Are there any ...
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Is there a name for this geometric property of metric spaces?
My research has lead me to metric spaces $(M, \rho)$ which have the following geometric property:
Suppose $x, y \in M$ and $r, s > 0$ such that
$(x, r) \neq (y, s)$,
$B[y; s] \subseteq B[x; r]$,
$...
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An inequality about metric spaces
I started studying this article(《$L^2$ CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE》) about 3 months ago: arxiv.org/abs/1605.05583
In this article, there is a seemingly simple assertion ...
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universal 0-dimensional separable metric subspaces
Let $\ \mathscr U:=(U\ \delta)\ $ be a separable metric space which is universal for all finite metric spaces, i.e. for every finite metric space $ \mathscr X:=(X\ d)\ $ there
exists an isometric ...
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Fractional Hajłasz-Besov-like similar to the Korevaar-Schoen-Sobolev spaces?
Suppose that $(X,\mu,d)$ and $(Y,\nu,\rho)$ are doubling metric measure spaces. Fix $\alpha>0$ and define the space, analogously to this paper, as the collection of all measurable functions $f:X\...
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Does rate of convergence in probability come from a metric?
In general, when we talk about convergence of a sequence, we need a topological space. If we want to talk about a rate of convergence, we need to quantify how far away one element of the sequence is ...
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Graph contained in a metric space
I have a metric space $X$ and a graph $G=(V,E)$ whose set of vertices is a subset $V\subset X$ (and $E$ is the set of edges, which is a symmetric subset of $V\times V$). For each $v\in V$, the set of ...
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Is the following product-like space a Polish space?
Let $\mathcal{M}_1(\mathbb R)$ denote the space of Borel probability measures on $\mathbb R$. The space is a Polish space (a space which admits a complete, separable, metric) using, say the Levy-...
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"monotone" homotopy?
This is a question about a concept that I call "monotone homotopy" which arises in a natural way in some topological situations.
Let $X$ be a (bounded) metric space, $Y$ be a topological space and $A\...
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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 ...
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There exists differentiable curves arbitrarily close to the continuous ones
Let $M$ be a Riemannian manifold; if $d$ is the distance on $M$, we can consider the distance $D$ between any two continuous curves given by $D(c, \gamma) = \max _{t \in [0,1]} d(c(t), \gamma(t))$.
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Metrization of spaces of functions
Let $M$ and $N$ be topological spaces. Are there necessary and sufficient conditions on the topological properties of the spaces such that $C(M,N)$ is metrizable?
For $M$ compact and $N$ a metric ...
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Compactness of sigma-algebra for the $L^1$ metrics
Consider a probability space $(X,F,\mu)$, and the quotient $G$ of the sigma-algebra $F$ by its null sets. Endow $G$ with the metric $d(A,B) = \mu(A \triangle B)$. Is $(G,d)$ a compact metric space?
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Two metrics and a sequence converging to two points. [closed]
Suppose I have a set with two metrics, which induce distinct topologies, (so neither is contained in the other). There should exist a sequence which converges in both topologies, but to different ...
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For $\mathbb R^n \times Q \cong \mathbb R^m \times Q $ must $n = m$? ($Q$ is the Hilbert cube)
There are several theorems describing the topology on hyperspaces of convex subsets of $\mathbb R^n$ under the Hausdorff metric. For example Antonyan and Jonard-Pérez prove the space of compact convex ...
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Linear process close to a Gaussian process
A linear process $(X_t)_{t \in \mathbb{Z}}$ is usually written as a moving-average process with infinity order:
\begin{equation}\label{linear_process}\tag{Eq. 1.1}
X_{t} = \sum_{j =0 }^\infty \...
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Stretching map of $n$ points from $\{0,1\}^n$ to $\{0,1\}^{n+1}$ with respect to their Hamming distance
Given a positive integer $n$, the Hamming distance $d^H_n(x,y)$ of $x,y\in \{0,1\}^n$ is defined by $$d^H_n(x,y) = |\{k\in\{0,\ldots,n-1\}: x(k)\neq y(k)\}|.$$
Given an integer $n>0$ and a set $S\...
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Under what conditions can we put a complete norm on a linear subspace of a separable Banach space?
Question 1 Let $X$ a separable Banach Space and $Y\subset X$ linear subspace. When can we put a norm on $Y$ in such a way so that $Y$ is a Banach space?
Clearly if $Y$ is closed in the norm topology ...
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Independence of the axiomatics of metric cones
A metric cone $C$ is a nonempty metric space (whose distance is denoted $d$) together with a map $\cdot\colon \mathbf{R}\times C \mapsto C$ satisfying these axioms:
$a\cdot(b\cdot x) = (ab)\cdot x$ ...
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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 ...
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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(\...
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Partitioning a smooth manifold into geodesically convex sets
Let $X$ be a connected and compact $d$-dimensional smooth manifold; where $d$ is a positive integer. Does (or rather, when does) there exist a metric $\rho$ on $X$ generating $X$'s topology and a ...
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"Geodesic coherent" partition of a graph
Let $G=(V,E)$ be a finite undirected graph which we equip with its usual graph geodesic distance $d_G$ making $(G,d_G)$ into a metric space; let $1<\#V<\infty$. For a given $1<N< \#V$ ...
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Path Metric Topology
Is there an example of a metric space $(X,d)$ whose corresponding path metric, $d^\prime$ generates a strictly finer topology compared to the topology generated by $d$?
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Are there any statistical metrics that satisfy this kind of condition?
Let $f=N(\mu,\sigma^2)$ be a univariate normal distribution with mean $\mu$ and variance $\sigma^2$ and let $f_1 = N(\mu+\epsilon,\sigma^2)$ and $f_2=N(\mu,(\sigma+\epsilon)^2)$ be some small ...
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Measures on complete metric spaces for which all meager sets are null
On a complete metric space the collection of meager and comeager sets form a $\sigma$-algebra. There is a 'natural' measure you can put on this $\sigma$-algebra where the measure of a meager set is 0 ...
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What is the most ``diverse'' $k$-subset of $[0, 1]^m$?
Given a non-negative integer $m$, let $\Omega_m$ denote the set of vectors $\omega = (\omega_1, \dots, \omega_m) \in [0, 1]^m$ such that $\sum_i{\omega_i} = 1$.
The set $\Omega_m$ is together with a ...
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Is this map (from the space of probability densities to the Wasserstein space) Lipschitz?
Let $p \in [1, \infty)$. Let $\mathcal P_p(\mathbb R^d)$ be the space of all Borel probability measures on $\mathbb R^d$ with finite $p$-th moments. Let $D_p$ be the collection of all Borel measurable ...
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Lipschitz-free space of countable uniformly discrete metric space
I assume here that the reader is familiar with the concept of Lipschitz-free space $\mathcal{F}(X)$ of a metric space $X$. I will follow the definition of $\mathcal{F}(X)$ as the completion of the ...
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Expected measure of a ball in a probability space with a metric
Assume we are given a probability space $(\mathbb{X}, \mathcal{X}, \mathbb Q)$ and a measurable distance function defined on it $d:\mathbb{X}\times \mathbb{X}\to \mathbb{R}^+\cup\{0\}$ that conforms ...
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Reference request: extendability of Lipschitz maps as a synthetic notion of curvature bounds
In the lecture Notions of Scalar Curvature - IAS around 8:00, Gromov states the following result, which he claims he does "slightly uncarefully":
Suppose $(X,g_X)$ and $(Y,g_Y)$ are ...
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Is this sum of nonexpansive maps itself nonexpansive?
For Hilbert spaces $\mathcal{H}_X$, $\mathcal{H}_Y$ and $\mathcal{K}$, consider a linear map $f\colon \mathcal{K} \oplus \mathcal{H}_X \to \mathcal{K} \oplus \mathcal{H}_Y$ that is given as a matrix
$...
3
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1
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607
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Quotient of compact metrizable space in Hausdorff space
Lets $X$ be a compact metrizable space and $f:X\to Y$ be a quotient map such that $Y$ equipped with the quotient topology is Hausdorff. Thus $Y$ is metrizable. Lets $\sim$ be an equivalence relation ...
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Ultraproduct of metric spaces
Let $I$ be a set and $\mathcal{U}$ be an ultrafilter on $I$. Suppose that $(X_{i}, d_{i})_{i\in I}$ is a family of pointed metric spaces with a distinguished point $e_{i}$ for each $i\in I$. We set
$$...
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When is a real-valued function on a metric space a metric?
Let $X$ be a space, $\ f:X\times X\rightarrow\mathbb{R}^+\cup\{0\}$ be a map satisfying the first two axioms for a metric (so $f(x,y)=0$ exactly when $x=y$, and $f$ is symmetric).
Now, consider the ...
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