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Questions tagged [metric-geometry]

The study of geometry of manifolds without appealing to differential calculus. It includes studies of length spaces, Alexandrov spaces, and CAT(k) spaces. The techniques are often applicable to Riemannian/Finsler geometry (where differential calculus is used) and geometric group theory. For questions about plain-old metric spaces, please use (metric-spaces) instead.

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Center of Carnot Group [closed]

I know the center of the Heisenberg group (over $\mathbb{R}$) is isomorphic to $\mathbb{R}$, but what about the wider class of Carnot groups? Are their centers also $\mathbb{R}$?
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Contractable Metric Spaces Homeomorphic to Euclidean Space

Is there a characterization of all metric spaces which are homeomorphic to a contractable subset of Euclidean space? This question is cross-referenced here.
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Existence of $1$-Lipschitz map between triangles

Consider two (Euclidean) triangles $T$ and $T'$. Let's say that $T$ majorizes $T'$ if there exists a 1-Lipschitz map that sends vertices to vertices and sides to sides (for some labeling of the ...
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1answer
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Limit of points in ellipse

Consider $$c(t)=(a\cos\ (t),b\sin\ (t)),\ 0\leq t,\ a=1 > b>0$$ in $\mathbb{R}^2$ Fix $0<t_1<\pi/2$ so that $x_1=c(t_1)$ When $x_i=c(t_i)$ and $x_i$ is a positive multiple of $c'(t_{i-...
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How is $\overline{c_{\epsilon}}:=\overline{\bigcup_{n\geq 1}\left(\bigcap_{k=n}^{\infty}B(x_k,r+\epsilon)\cap K \right)}$ convex in a CAT$(0)$ space?

Let $(X,d)$ be a CAT$(0)$ space, $\{x_n\}\subset X$ be bounded and $K\subset X$ be closed and convex. Define $\varphi:\,X\longrightarrow \mathbb{R},$ by $\varphi(x)=\limsup\limits_{n\to\infty}d(x,x_n)$...
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Duality and bounds on Lebesgue measure

A nice exercise was posted yesterday on Google+ : given the ellipse of equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, find the rectangle with maximal area whose corners lie on this ellipse. I realized ...
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Length is invariant under parameterization

I'm reading the book A Course in Metric Geometry by Dmitri Burago, Yuri Burago Sergei Ivanov. In page 45 it says "it is easy to see that all parametrization of a curve have equal length", but I do not ...
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Exact sequence in cohomology while studying orientability of Alexandrov spaces

I'm reading the article Orientability and fundamental classes of Alexandrov spaces with applications by Ayato Mitsuishi. There are the following definitions: for $n\geq 1$, an $n$-dimensional MCS ...
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1answer
102 views

Why the length is lower semi-continuous?

I'm reading the book A Course in Metric Geometry by Dmitri Burago, Yuri Burago Sergei Ivanov and I don't understand the proof of Proposition 2.5.17 (Page 48). More precisely, I don't know why the ...
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Every finite metric space $X$ can be embedded into $\ell_{C_\epsilon\log(|X|+1)}$ with low distortion

I have been tasked with proving that for all $\epsilon >0$ there exists some $C_\epsilon$ such that every finite metric space $X$ embeds into $\ell_{C_\epsilon\log(|X|+1)}$ with distortion at most $...
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Reference request: Introduction to Finsler manifolds from the metric geometry point of view (possibly from the Busemann's approach)

I was reading about geometry in metric spaces from different books, two of them are: (1) A course in metric geometry by Y. Burago, D. Burago and S. Ivanov; and (2) Metric spaces of non-positive ...
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1answer
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Free faces in $M_\kappa$ polyhedral complexes.

I am reading Bridson and Haefliger's ``Metric Spaces of Non-Positive Curvature" and I am struggling with Definition 5.9 of a free face. Let $K$ be an $M_\kappa$-polyhedral complex. A closed $n$-...
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1answer
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Metric space analog for manifold with boundary

I tried to define an analog of manifold with\without boundary for metric space and I wondered if this definition or a similar one exists in the literature. Let $\left(X,d\right)$ be a connected ...
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Finite number of jumps in metric for space with finite number of components

Let $(X,d)$ be a metric space with at most countably-infite many discontinuities and fix $x \in X$. For every $r \geq 0$, define the sets $A(r)$ by $$ A(r)\triangleq \{y \in X : d(x,y)=r\}, $$ and ...
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Every riemannian length structure on $\mathbb{R}^n$ is induced by a continuous function $f:\mathbb{R}^n\to \mathbb{E}^n$, to the euclidean space.

I have been reading about length spaces in the (great) book Metric Geometry by Y. Burago, D. Burago and S. Ivanov. They define what an induced length structure is and they do the following claim ...
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In what sense is an ultralimit a “limit”

Recently, I ended up having to use the notion of an ultralimit of metric spaces and realized that I do not have a good intuition for the "limit object" this construction creates. Given a sequence of ...
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1answer
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Request for applications of the Filling Theorem: geometric and algebraic Dehn functions are asymptotically equivalent

I am studying geometric group theory and metric spaces of bounded curvature. In particular, I was reading the seminal Gromov's Hyperbolic Groups and trying to filling the details. In the first page, ...
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Geodesics meeting with angle $0$ in ${\rm CAT}(0)$ space

Consider two distinct geodesics $\gamma_1$ and $\gamma_2$ in a CAT($0$) space, issued from the same base point. A trivial example where we have $\angle(\gamma_1, \gamma_2)=0$ is when $\gamma_1(t) = \...
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Isometry in a compact space is surjective

Let's say we have a compact metric space $(X,d)$ and a function $f: X\to X$ satisfying $d(f(x),f(y)) = d(x,y)$, $x,y\in X$. How can we show that this function is sujective in a simple way.
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bottle neck distance: distance to diagonal points

recall the bottleneck distance is defined as the minimum of max_x |x-f(x)| over all bijections f between points in persistence diagram A and persistence diagram B. Recall we include all diagonal ...
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Some hypersurface has a positive second fundamental form potentially

Notation : $r^2=x^2+y^2$ Exercise : Define $$F_\sigma (x,y)= (f_\sigma (x,y),\frac{x^2-y^2}{2},xy)$$ where $$f_\sigma (x,y)=(1- \sigma^2 r^2)(x-\sigma x^3,y-\sigma y^3)$$ Define $ G_\sigma: \mathbb{...
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Is finiteness of Assouad dimension a topological invariant for compact metric spaces?

A metric space is called doubling if there is some $C>0$ such that for any $r>0$ any ball of radius $r$ can be covered by $C$ balls of radius $r/2$. This is equivalent to having finite so-...
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1answer
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Dual element of supporting function

Problem : $(\mathbb{R}^n,\|\ \|)$ has a smooth and strictly convex norm. When $f(x)=\|x\|$, then find a directional derivative of a function $f$, i.e. $\frac{d}{dt}f(x+tv)$ for $\|x\|=1$. Refer : [1] ...
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Converting polar coordinates (degree) to Cartesian line coordinates

I am trying to calculate the (x1,y1), (x2,y2) coordinates of a line. From the image consider the rectangle of width w, height h, center c and angle θ. If the same is given in a graph with X and Y ...
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1answer
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Angles in CAT$(\kappa)$ spaces

My question is about the first two pages of Chapter II.3 of Bridson/Haefliger. Below $M^2_{\kappa}$ refers to the model 2-dimensional plane of curvature $\kappa$ (e.g., $S^2$ for $\kappa=1$ or $H^2$ ...
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1answer
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Product of two CAT($\kappa$) spaces is CAT($\kappa$) for $\kappa \ge 0$

I would like to see a "metric proof" that if two metric spaces $X$ and $Y$ are CAT($\kappa$) for some $\kappa \ge 0$, then so is their product. I would be satisfied to see a proof for $X=Y=S^2$. By "...
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1answer
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Approximation of four dimensional ball by three dimensional spheres

Question : There is a sequence of Riemannian metrics on 3-dimensional sphere s.t. they converges to four dimensional ball in Gromov-Hausdorff $d_{GH}$ sense. How can we prove this ? (cf. 105p. in [...
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An identity is generated from inscribed center of a simplex

I'm looking for a proof for this problem on simplex which I am sure it is true Question. Let $\mathcal{A}=A_0A_1...A_n$ be a simplex in $\Bbb E^n$. $I$ is the center of its inscribed sphere. Denote ...
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1answer
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Converses of Ptolemy's theorem in n-dimensional space

In Euclidean $n$-dimensional space $\Bbb E^n.$ We consider four points $A,$ $B,$ $C,$ $D.$ My question. Assume that $$d(A,B)\cdot d(C,D)+d(A,D)\cdot d(B,C)=d(A,C)\cdot d(B,D)$$ then four points $A,...
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Length of a curve defined by a convex function [closed]

Let $f \colon [0,1] \to [0,1]$ be a function of class $C^1$ such that $f(0)=f(1)=1$ and $f'$ is nondecreasing, i.e., $f$ is convex. Show that the length of the curve defined by the graph of $f$ is ...
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Combinatorial property of cross polytopes [duplicate]

I apologize in advance; I know that this site is for research-level mathematics, not for elementary learning ground. But I tried to understand the following question on my own but it is still ...
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$2$-dim positively curved Alexandrov space

A statement in the book I am reading states that any $2$-dimensional positively curved closed Alexandrov space is homeomorphic to $S^2$ or $\mathbb RP^2$. Is there any reference for this fact?
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1answer
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Formalize idea of 'homeomorphism that preserves geodesics'?

I am seeking to formalize the following idea: Take a sheet of paper and lay it flat on a table. Choose two points A and B on the paper, then draw a line segment from A to B. Fold the sheet of paper so ...
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Spaces of separable metric spaces II: pointed spaces

In the first question, Spaces of (complete) separable metric spaces, I asked about the Gromov–Hausdorff metric. Here, I am asking about the weaker notion of Gromov–Hausdorff convergence for pointed ...
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1answer
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Spaces of (complete) separable metric spaces

First, I want to make sure that I've got something right. By my understanding: There are separable (complete) metric spaces which are universal, i.e. contain isometrically embedded copies of every ...
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exercise about doubling

In Euclidean space, any $R$-ball can be covered by $C$ $\frac{R}{2}$-balls where $C$ is independent of $R$. But in hyperbolic space, this does not hold. Here we have a definition : Def : A ...
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1answer
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Example of length structure with Euclidian intrinsic metric but different path lengths

This is an exercise from "A course in metric geometry" by Burago and Ivanov I am having some troubles with. Give an example of length structure on the plane for which all continuous curves are ...
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1answer
108 views

Why not define lines in a metric space using a locus?

If we want to talk about geometry in a metric space, we go through the following procedure to define geodesics. We define the length of a path in the (complete) space as the length given by ...
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1answer
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Kuratowski convergence for unbounded/non-compact sets

Definition of Kuratowski convergence as copied from the Wikipedia page: Let $(X,d)$ be a metric space, where $X$ is a set and $d$ is the function of distance between points in $X$. For any $x \in X$ ...
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Angles made by a segment in two concentric circles

I have a quite tricky geometry problem to solve, and I don't find the answer. Here is the problem: I have two concentric circles of known radii (r for the inner circle and R for the outer one). ...
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Riemann Metric of $\mathcal M_1\cup\mathcal M_2$

Suppose $\mathcal M_{1},\ \mathcal{M}_{2}$ are $(n-1)$-manifold embedded in $\mathbb R^n$. $d_i$ is the geodesic distance defined on $\mathcal M_i$. What is the name of the distance and metric ...
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1answer
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Second derivative of distance function to a point in model spaces.

I'm wondering if someone could please point me out to a reference (or the actual identity) where the following is shown. Let $\gamma:I\to (\mathbb{R}^n,d=d_{euclidean})$ be a geodesic in $n$-...
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1answer
51 views

Restricting Foliations by Hopf Circles

Suppose I have $\mathbb{S}^5$, foliated by Hopf circles. I am wondering if this restricts in some way to the foliation by Hopf circles on $\mathbb{S}^3$ in the join $\mathbb{S}^5=\mathbb{S}^3*\mathbb{...
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1answer
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Is the notion of 'divergence to infinity in a direction' used?

In $\mathbb R$ a sequence can diverge to infinity in two directions: $+\infty$ and $-\infty$. These two cases of divergence are quite different from a sequence that diverges to "nowhere", like $\{(-1)^...
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1answer
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Does Gromov-Hausdorff distance depends on the choice of the “ambient” metric space?

Give two metric space $X,Y$, we denote by $d_{GH}(X,Y)$ the Gromov- Haudorff distance. It seems that it only relies on $X,Y$ not on the ambient metric space $Z$ where we embed $X,Y$ (because in the ...
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1answer
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What information can be recovered from the induced metric of a Riemannian manifold?

Note: In what follows, "metric" means "metric space metric", not "Riemannian metric". Imagine you are playing a game with a friend. You choose a Riemannian manifold $M$, thus a point set $M$ with an ...
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Does curvature affect these “infinitesimal isometry groups”?

This is a follow-up to a previous question, where it was concluded that at any point of Euclidean space, the "infinitesimal isometry group" is $O(n) \times \mathbb{R}^n$ and the "pointed infinitesimal ...
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1answer
227 views

Do local isometries form a group? If so, what is the group for Euclidean space?

Burago, Burago, and Ivanov use the following definition on p.78, for metric spaces $(X,d)$, $(Y,\delta)$: A map $f: X \to Y$ is called a local isometry at $x \in X$ if $x$ has a neighborhood $U_x$ ...
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1answer
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When does a group of dilations/scalings exist in a metric space?

Notation: Let $(X,d)$ be a metric space. A similitude will be (by convention) a surjective (hence bijective) map $f: X \to X$ such that for all $x_1, x_2 \in X$, $d(f(x_1),f(x_2)) = r d(x_1, x_2)$ for ...
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1answer
86 views

Are small balls in a metric space quasi-symmetric to a compact ball connected?

I'm trying to solve a problem but I'm stuck. Maybe someone can help me. I will denote every metric on every metric space by $d$ and I will use closed balls, denoting them with the letter $B$, so for $...