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.

Filter by
Sorted by
Tagged with
0
votes
0answers
24 views

Rolling a 3-dimensional compact convex body on a flat plane.

Consider $\Sigma$, which is $2$-dimensional smooth Riemannian manifold in $\mathbb{R}^3$ of positive Gaussian curvature homeomorphic to $\mathbb{S}^2$. Assume that $\alpha :[0,\infty)\rightarrow \...
1
vote
0answers
8 views

$G$-invariant metrics on homogeneous spaces $G/H$

Let $G$ be a compact metrizable group and let $H$ be a closed subgroup. Let $d$ be a compatible metric on $G$ that is left-invariant and it is also right invariant with respect to the elements of $H$. ...
0
votes
0answers
13 views

Volume of the intersection of $n+1$ isometric $n$-balls that radius is the distance betwin any two of their center

Let $B_0, B_1,...,B_n$ be $n$-balls of radius $1$ such that for each of theses $n$-balls the center of any other in his edge. Let's call $r_n$ the radius of an $n$-ball that has the same volume as ...
1
vote
0answers
30 views

Pointwise convergence is uniform convergence under a condition

[1] pointwise convergence is uniform convergence under some condition [2] A course in metric geometry - Burago, Burago and Ivanov Question : I want to understand intuitively or visually the proof ...
1
vote
1answer
41 views

pointwise convergence is uniform convergence under some condition

When $f : \mathbb{R}^2\rightarrow \mathbb{R}$ is a nonnegative continuous function, then define $F (v) = \lim_{t\rightarrow \infty}\ \frac{f(tv)}{t}$. When $F$ is a norm on the vector space $\mathbb{R}...
0
votes
0answers
30 views

Can invariant metrics on quotients of groups $G/H$ always be obtained from invariant metrics on $G$?

Let $d$ be a (left-)invariant and compatible metric on a compact group $G$. Let $H$ be a closed subgroup of $G$. Then one can define on the homogeneous space $G/H$ an invariant and compatible metric $...
0
votes
1answer
41 views

Does convexity implies contractibility in length space?

It is known that convexity implies contractibility in Euclidean spaces. I want to know whether it holds in a general length space. To be specific: Let $(X,d)$ be a length space and $A\subseteq X$. $A$...
2
votes
0answers
49 views

When is the quotient of a geodesic space again a geodesic space?

I am interested in the behavior of the quotient semi-metric on geodesic spaces, i.e. length spaces where there is always a minimal curve between two points. I used the following definition of the ...
0
votes
0answers
65 views

Completion of Length Space is Length Space

In Burago, Burago, and Ivanov’s “A Course in Metric Geometry” you are asked (exercise 2.4.18) to show that the completion (Y, e) of a length space (X, d) is a length space. I feel like this is fairly ...
3
votes
1answer
101 views

what are the inner angles of a tetragonal trapezohedron

I am making a 3D wooden lantern in the shape of a Tetragonal Trapezohedron. Think of a hollow Tetragonal Trapezohedron 3D object, where each face is a plank of wood. I am trying to determine the ...
2
votes
0answers
114 views

Example of a quasi-open set and how the capacity of the set fits in it?

While going through some research papers, I came across a result of D. Bucur where the existence of a minimizer for the general $k^{th}$ eigenvalue of the Dirichlet Laplacian among a class of quasi-...
1
vote
1answer
50 views

Length of the diagonal of a “dented triangle”

Consider a "dented triangle" like in this picture: (the angle $\alpha$ is larger than $\pi$). Compare it with an actual triangle $\triangle\bar{A}\bar{B}\bar{C}$ with sidelengths $a, b$ and $c = c' + ...
1
vote
1answer
100 views

Understanding a Proof: The square root of any metric is ptolemaic..

In Hyperbolicity, CAT(-1)-spaces and the Ptolemy Inequality, there is a short proof of a simple statement: Let $(X, d)$ be an arbitrary metric space. Then $(X, \sqrt{d})$ satisfies the Ptolemy ...
1
vote
1answer
58 views

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.
3
votes
2answers
97 views

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 ...
0
votes
1answer
41 views

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-...
1
vote
1answer
88 views

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 $\varphi(x)=\limsup\limits_{n\to\infty}d(x,x_n)$...
2
votes
0answers
93 views

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 ...
2
votes
0answers
77 views

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 ...
2
votes
1answer
164 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 ...
1
vote
0answers
36 views

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 $...
3
votes
2answers
144 views

Reference request: Introduction to Finsler manifolds from the metric geometry point of view (possibly from the Busemann's approach)

This question is a cross post from MathOverflow. I have requested the migration of the question, but unfortunately it is not possible after two months of posting. I was reading about geometry in ...
0
votes
1answer
29 views

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 $n$-...
1
vote
1answer
51 views

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 ...
1
vote
0answers
22 views

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 ...
3
votes
1answer
154 views

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.

This question is a cross post from MathOverflow. Unfortunately the migration of the question is not possible after two months of posting. I have been reading about length spaces in the (great) book ...
3
votes
0answers
138 views

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 ...
1
vote
1answer
188 views

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, ...
5
votes
2answers
149 views

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) = \...
1
vote
2answers
197 views

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.
0
votes
1answer
66 views

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 ...
1
vote
0answers
114 views

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{...
1
vote
0answers
71 views

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-...
0
votes
1answer
90 views

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] ...
0
votes
2answers
92 views

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 ...
1
vote
1answer
58 views

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 $H^2$ ...
3
votes
1answer
70 views

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 "...
0
votes
1answer
151 views

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 [...
6
votes
2answers
187 views

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 ...
1
vote
0answers
25 views

$2$-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?
2
votes
1answer
158 views

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 ...
2
votes
0answers
51 views

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 ...
2
votes
1answer
255 views

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 ...
0
votes
0answers
30 views

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 ...
2
votes
1answer
101 views

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 ...
6
votes
1answer
127 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 ...
0
votes
1answer
114 views

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 $x \in X$ ...
1
vote
0answers
237 views

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). ...
0
votes
0answers
42 views

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 ...
2
votes
1answer
125 views

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 $n$-...