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Metric for measuring linearity of finite set of points in $R^2$

Suppose one has $n > 2$ points in $R^2$, and one wants to measure "how linear" they are. I want a metric such that (a) if all the points are in fact on the same line, the metric gives 1, (...
Michael Mc Gettrick's user avatar
4 votes
1 answer
97 views

Inner regularity property of covering number of metric spaces

Let $(X,d)$ be a complete metric space and $n\in\mathbb N$. Suppose that every finite subset $F\subset X$ can be covered by $n$ closed balls of $X$ (that is, $N(Y,d,1)\le n$, in terms of covering ...
Pietro Majer's user avatar
  • 60.5k
1 vote
0 answers
33 views

Obtaining the geodesic extension property by embedding in a larger space

Suppose $(X,d)$ is a Hadamard space. By considering basic examples like a compact interval in $\mathbb{R}$ or a closed unit ball in Hilbert space, $X$ need not have the geodesic extension property (...
E G's user avatar
  • 163
0 votes
0 answers
37 views

Does smallness of Gromov-Hausdorff distance on scale 2 imply smallness on GH distance on scale 1?

Let $(M,g)$ be a Riemannian manifold and $C(Y)$ be a metric cone over $Y$. Let $B_r$ denote the geodesic ball of radius $r$ centered at a fixed point $x$ in $M$ and $C_r$ denote the metric ball of ...
Y.Guo's user avatar
  • 151
8 votes
0 answers
149 views

Do the $\ell^{\infty}$ and $\ell^1$ norms yield minimal doubling constants amongst all norms on $\mathbb{R}^n$?

Setting: Let $X:=\mathbb{R}^n$ for some positive integer $n$. For each $1\le p\le \infty$ let $d_p$ denote the metric induced by the $\ell^p_n$ norm thereon. Note that, the doubling constant of a ...
ABIM's user avatar
  • 5,405
5 votes
1 answer
483 views

Can you always extend an isometry of a subset of a Hilbert Space to the whole space?

I remember that I read somewhere that the following theorem is true: Let $A\subseteq H$ be a subset of a real Hilbert space $H$ and let $f : A \to A$ be a distance-preserving bijection, i.e. a ...
Cosine's user avatar
  • 609
1 vote
0 answers
42 views

Genaralizing the metric expression present in the quadrilateral inequality

Let $(X, d)$ be a metric space. In Sato - An alternative proof of Berg and Nikolaev’s characterization of CAT(0)-spaces via quadrilateral inequality it is stated that if $X$ is a geodesic space, then ...
Kacper Kurowski's user avatar
0 votes
0 answers
77 views

Wasserstein space isomorphic to original space?

Is there a complete measurable metric space $(X,d)$ for which its $p$-Wasserstein space $W(X)$ is isometrically isomorphic to $(X,d)$ for some $p \in [1,\infty]$? Note that there is a canonical non-...
Florentin Münch's user avatar
-2 votes
1 answer
141 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 ...
mathoverflowUser's user avatar
6 votes
0 answers
184 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-...
David Bryant's user avatar
3 votes
1 answer
161 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(\...
user1598's user avatar
  • 177
0 votes
1 answer
94 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?
Samir's user avatar
  • 43
1 vote
1 answer
161 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 ...
Strichcoder's user avatar
2 votes
2 answers
226 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 $x,y,z \in X$, ...
gm01's user avatar
  • 327
1 vote
1 answer
116 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 ...
Math_Newbie's user avatar
0 votes
1 answer
410 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 $...
C L 's user avatar
  • 101
4 votes
1 answer
210 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{...
ABIM's user avatar
  • 5,405
2 votes
1 answer
46 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 ...
pyridoxal_trigeminus's user avatar
1 vote
0 answers
126 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 ...
Bedovlat's user avatar
  • 1,959
3 votes
0 answers
61 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 ...
Kacper Kurowski's user avatar
0 votes
1 answer
131 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 ...
Justin_other_PhD's user avatar
0 votes
1 answer
115 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 ...
Justin_other_PhD's user avatar
1 vote
0 answers
125 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 ...
Justin_other_PhD's user avatar
1 vote
1 answer
276 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 ...
Kacper Kurowski's user avatar
3 votes
1 answer
132 views

If $X,X'$ have the same $\varepsilon$-packing numbers and $f:X \to X'$ surjective $1$-Lipschitz, then $f$ is an isometry

Let $(X, d)$ be a compact metric space. We say that $\{x_1, \cdots, x_n\} \subseteq X$ is an $\varepsilon$-covering of $X$ if for any $x \in X$, there exists $i \in \{1, \ldots, n\}$ such that $d(x, ...
Akira's user avatar
  • 835
2 votes
1 answer
139 views

Are two metric spaces isometric if they have the same $\varepsilon$-covering and $\varepsilon$-packing numbers for all $\varepsilon>0$?

Let $(X, d)$ be a compact metric space. We say that $\{x_1, \cdots, x_n\} \subseteq X$ is an $\varepsilon$-covering of $X$ if for any $x \in X$, there exists $i \in \{1, \ldots, n\}$ such that $d(x, ...
Akira's user avatar
  • 835
2 votes
1 answer
259 views

Are two metric spaces isometric if they have the same $\varepsilon$-covering numbers for all $\varepsilon>0$?

Let $(E, d)$ be a metric space. For $\varepsilon>0$, we define two notions of $\varepsilon$-covering number as follows, i.e., $N_\varepsilon^o (E)$ is the smallest number of open balls whose radii ...
Akira's user avatar
  • 835
4 votes
1 answer
292 views

Is every 1-Lipschitz homeomorphism $f:X\to X$ from a compact metric space to itself an isometry?

I found a statement involving a homeomorphism $f:X\to X$ of a compact metric space $X$, with Lipshitz coefficient 1, i.e., a non-expansive map, and cannot think of an example where $f$ is not an ...
Saúl RM's user avatar
  • 10.6k
1 vote
0 answers
165 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 ...
user243245's user avatar
2 votes
1 answer
110 views

Lipschitz maps with Hölder inverse preserve the doubling property

Let $K$ be a compact doubling metric space, $X$ be a metric space and $f:K\rightarrow X$ be Lipschitz with $\alpha$-Hölder inverse, where $0<\alpha<1$. Does $f(K)$ need to be doubling?
ABIM's user avatar
  • 5,405
1 vote
1 answer
164 views

Right-continuity of covering number

Consider an ambient metric space $(\mathcal{X},\Vert\cdot\Vert_\infty)$. Let $\mathcal{B}_1 = \mathcal{B}_{\Vert\cdot\Vert_K}(0,1)\subseteq\mathcal{X}$ be the closed unit ball with respect to some ...
iom10's user avatar
  • 23
1 vote
1 answer
221 views

What properties are preserved by quasi-isometries

Recently, I came across the notion of quasi-isometries, while thinking of "discrete spaces which are surrogates for approximate continuous ones". What (metric)/geometric properties are ...
ABIM's user avatar
  • 5,405
3 votes
1 answer
107 views

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 ...
pyridoxal_trigeminus's user avatar
6 votes
1 answer
257 views

Expected doubling constant of a random Erdős–Rényi graph

Consider the $G(n,p)$ random graph model where $n$ is a ``large'' positive integer and $p\in (0,1)$. We may equip every realized random graph $G$ with its shortest path distance, making it into a (...
ABIM's user avatar
  • 5,405
6 votes
1 answer
284 views

Extending a partially defined metric on a metrizable space

Let $X$ be a metrizable topological space, $A\subseteq X\times X$ a nonempty closed subset which is reflexive, symmetric and transitive, $d:A\to \mathbb{R}_+$ a continuous function that satisfies the ...
omar's user avatar
  • 278
6 votes
1 answer
237 views

m-point-homogeneous, but not (m+1)-point-homogeneous

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 $A\subset Q$ ...
Anton Petrunin's user avatar
4 votes
1 answer
159 views

Extending a metric in a bi-Lipschitz way

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 ...
an_ordinary_mathematician's user avatar
0 votes
1 answer
61 views

$\omega$-homogenous space which is not $\omega_1$-homogenous

Consider a metric space $(X,d)$ and let $\kappa$ be a cardinal. We say that $(X,d)$ is $\kappa$-homogenous, if every (surjective) isometry $h:X_1 \to X_2$ between subspaces of $(X,d)$ of size $< \...
Reijo Jaakkola's user avatar
3 votes
1 answer
244 views

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 ...
ABIM's user avatar
  • 5,405
3 votes
1 answer
135 views

"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 $1<N< \#V$ ...
ABIM's user avatar
  • 5,405
2 votes
0 answers
115 views

Definition of the category QMet of metric spaces and quasi-isometries

I am following Clara Löh's Geometric Group Theory. An Introduction, and in remark 5.1.12, she defines the category QMet whose objects are metric spaces and whose morphisms are quasi-isometric ...
Saúl RM's user avatar
  • 10.6k
2 votes
2 answers
231 views

$(1+\epsilon)$-bilipschitz parametrization of Lipschitz manifold

Let $\mathscr{H}^m$ be the $m$ dimensional Hausdorff measure in $\mathbb{R}^n$, $m\leq n$. Is it true that for $\mathscr{H}^m$-almost every point $p$ on a Lipschitz manifold $M$ of dimension $m$ ...
No-one's user avatar
  • 1,149
3 votes
0 answers
158 views

Constant in Naor and Neiman's Assouad Theorem

In Naor and Neiman's Assouad embedding theorem - "Assouad’s theorem with dimension independent of the snowflaking" Revisita Mathematica, the authors derive quantitative estimates on the ...
ABIM's user avatar
  • 5,405
3 votes
0 answers
115 views

Isometric embeddings of $c_0$ into metric spaces

Are there any nice and useful criteria or theorems which assert when a given metric space $M$ contains an isometric (not necessarily linear) copy of the Banach space $c_0$ or its unit ball $B_{c_0}$? (...
Damian Sobota's user avatar
0 votes
1 answer
514 views

Distance between two points using triangulation

Suppose we have two points $p_1$ and $p_2$ in a metric space with unknown dimensionality, with no way to directly compute the distance between them, e.g. no coordinates. Say we can randomly sample a ...
CambridgeStudent's user avatar
0 votes
0 answers
49 views

When can a compact metric space be covered by finitely many nearly-disjoint closed and convex sets?

This question is a follow-up of the following negative question. Let $(X,d)$ be a (non-empty) compact metric space. More generally than in the first post, I'll call a set of non-empty subsets $C_1,\...
ABIM's user avatar
  • 5,405
0 votes
0 answers
62 views

Terminology: maps which are bi-Lipschitz on compact subsets

Let $X$ and $Y$ be metric spaces and let $f:X\rightarrow Y$ be such that: for every compact subset $K$ of $X$ the restricted map $f|_K:K\rightarrow Y$ defined by $f|_K(x)=f(x)$ is bi-Lipschitz (with ...
ABIM's user avatar
  • 5,405
4 votes
1 answer
407 views

Lipschitz-regularity of partition of unity

Let $K$ be a compact subset of $\mathbb{R}^n$ and $\mathcal{U}$ be a finite collection of open subsets covering $K$ satisfying the minimality property: for every $U\in \mathcal{U}$, the sub-collection ...
Carlos_Petterson's user avatar
1 vote
0 answers
238 views

Example of CAT($k$) space [closed]

Good time of day. I repeat the question from MSE (https://math.stackexchange.com/questions/4464888/question-about-example-of-catk-space) because no response has been received.Question is the following:...
UserIn's user avatar
  • 103
1 vote
0 answers
449 views

Bound on covering number of Lipschitz functions – missing part in proofs of Kolmogorov et al

Given a metric space $(\mathcal{X},\rho)$ and $\mathcal{A}\subset\mathcal{X}$ totally bounded, i.e. $\mathcal{A}$ has a finite $\varepsilon$-covering for any $\varepsilon>0$. Consider $\...
samuel's user avatar
  • 11