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.
Questions tagged [metric-geometry]
314 questions
25
votes
4 answers
Finding an invisible circle by drawing another line
A friend of mine taught me the following question. He said he found it in a book a few years ago. Though I've tried to solve it, I'm facing difficulty.
Question: You know on a plane there is an invisible circle whose radius is less than or equals…
mathlove
- 151,597
20
votes
1 answer
Is there an explicit left invariant metric on the general linear group?
Let $\operatorname{GL}_n^+$ be the group of real invertible matrices with positive determinant.
Can we construct an explicit formula for a metric on $\operatorname{GL}_n^+$ which is left-invariant, i.e.
$$d(A,B)=d(gA,gB) \, \,\forall A,B,g…
Asaf Shachar
- 25,967
18
votes
2 answers
Isometry group of a norm is always contained in some Isometry group of an inner product?
$\newcommand{\<}{\langle} \newcommand{\>}{\rangle} $Let $||\cdot||$ be a norm on a finite dimensional real vector space $V$.
Does there always exist some inner product $\<,\>$ on $V$ such that $\text{ISO}(|| \cdot ||)\subseteq \text{ISO}(\<,\>)$ ?…
Asaf Shachar
- 25,967
17
votes
1 answer
Gromov-Hausdorff convergence to a circle
I am working on the book A course in metric geometry written by D. Burago, Y. Burago and S. Ivanov, and more precisely on exercice 7.5.9:
Exercice: Let $\{X_n\}$ be a sequence of compact length spaces, $X_n \underset{GH}{\to} S^1$. Prove that, for…
Seirios
- 34,083
17
votes
2 answers
How to show that $\mathbb R^n$ with the $1$-norm is not isometric to $\mathbb R^n$ with the infinity norm for $n>2$?
Could you please give me a hint to prove that
$\mathbb{R}^n$ with the 1-norm $\lvert x\rvert_1=\lvert x_1\rvert+\cdots+\lvert x_n\rvert$ is not isometric to $\mathbb{R}^n$ with the infinity-norm $\lvert x\rvert_\infty = \max_{i=1,\ldots,n}(\lvert…
zerhan
- 171
- 1
- 4
15
votes
0 answers
Gromov-Hausdorff distance between a line segment and a cylinder
I want to prove the following statement, where $d_{GH}$ denotes the Gromov-Hausdorff distance:
For $X= S^1 \times [0,1]$ and $Y= [0,1]$ with the intrinsic metrics one has
$d_{GH}(X,Y) = \frac{\pi}{2} $
What I found in the literature is the…
asterisk
- 396
13
votes
6 answers
Can every finite metric space be approximated by a distinct distance space?
Call a finite metric space $(X,d_X)$ a distinct distance space if all the nonzero distances in $X$ are distinct, i.e., the set of nonzero distances has size exactly ${|X|\choose 2}$.
The answer to the following question has implications for my…
R. H. Vellstra
- 3,263
13
votes
2 answers
Metric Isometry is always smooth?
Let $M$ be a smooth manifold. Let $d$ be any metric on $M$ which induces the topology on $M$. Let $f:(M,d) \rightarrow (M,d) $ be an isometry (in the sense of metric spaces). Is it true that $f$ must be smooth?
(if the metric $d$ is induced by some…
Asaf Shachar
- 25,967
12
votes
1 answer
Lower semi-continuity of one dimensional Hausdorff measure under Hausdorff convergence
Let $\mathcal H^1$ be the one-dimensional Hausdorff measure on $
\mathbb R^n$, and let $d_H$ be the Hausdorff metric on compact subsets of $\mathbb R^n$. If $K_n$ is connected for all $n \in \mathbb N$, and $d(K_n,K) \to 0$, I would like to know if…
Justthisguy
- 1,601
11
votes
2 answers
Why is the Barycenter operation in Hadamard spaces Lipschitz continuous?
I am looking into exercise 9.2.22 of "A course in metric geometry" by Burago-Burago-Ivanov.
For a Hadamard space $H$ (a complete simply connected metric space of nonpositive curvature in the Alexandrov sense)
and a fixed point $p \in H$ the…
Near
- 556
10
votes
1 answer
isometries of the sphere
There is a theorem by Pogorelov that if a $C^2$ surface $M$ in $\mathbb{R}^3$ is isometric to the unit 2-sphere, then $M$ is itself (a rigid motion of) the sphere.
What is known about isometric deformations of the sphere, when the smoothness…
user7530
- 50,625
10
votes
1 answer
Quantifying the angle metric on the Grassmannian in terms of the norm on the exterior power
Let $V$ be a finite-dimensional Hilbert space and $Gr_k(V)$ the Grassmannian of $k$-dimensional subspaces of $V$.
The $k$th exterior power $\bigwedge^k(V)$ can be equipped with a scalar product by extending
$$
\langle v_1\wedge \cdots \wedge v_k,…
Sven Pistre
- 1,390
- 9
- 24
9
votes
2 answers
What is the mathematical motivation behind metric geometry
Sorry if the title of this question is not formatted in a way that stackexchange likes, though my question is actually much more specific.
I was reading the book A Course in Metric Geometry, by Burago, Burago, and Ivanov. So I understand the basic…
krishnab
- 2,653
9
votes
1 answer
Do any of these concepts relating to similarity in general metric spaces have names?
Definition Let a metric space $(X,d)$ be given. A similitude is a function $f: X \to X$ such that, for all $x,y \in X$, $d(f(x),f(y)) = r \cdot d(x,y)$ for some positive real number $r>0$.
Questions:
1. What is the name of metric space in which…
Chill2Macht
- 22,055
- 10
- 67
- 178
9
votes
2 answers
Set of diameter $\le 1$ contained in set of constant width $1$
I'm reading the paper Minimal universal covers in $E^n$ by H.G. Eggleston and they state that every set $A\subseteq{\bf R}^2$ of diameter at most $1$ (the diameter of $A$ is defined as $\sup_{x,y\in A}|x-y|$) is contained in a set which has width…
fweth
- 3,800