# 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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### 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 ...
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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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### 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 ...
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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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### 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.

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 ...
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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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### 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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### 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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### 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 :  ...
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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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### 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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### 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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### 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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### 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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### $2$-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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### 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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### 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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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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### 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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### 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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### 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). ...
### 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 ...
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 $n$-...