# 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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### 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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### 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.

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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### 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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### 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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