All Questions
Tagged with big-list mg.metric-geometry
10 questions
35
votes
17
answers
3k
views
Equivalent definitions of Gromov hyperbolicity
Let $X$ be a metric space. I'd like to collect as many definitions of Gromov hyperbolicity or $\delta$-hyperbolicity of $X$ as possible.
I'm happy for the definitions to require some niceness ...
2
votes
0
answers
146
views
What practically computable homotopy and/or (co)homology theories are known for finite (di)graphs, metric spaces, etc?
Of late I have taken to applying Dowker homology and the path homology theory of Grigor'yan et al. like a hammer to various relations and/or digraphs that have looked like nails. At the same time, I ...
8
votes
2
answers
850
views
Geometric or topological results from group theory
Do you know interesting examples of purely geometric or topological results which can be proved using group theory? To make precise what I have in mind, let us consider the two following examples:
...
2
votes
1
answer
614
views
Distance matrices
We say that a matrix $M\in\mathbb{R}^{n\times n}$ is a distance matrix on a metric space $(X,d)$, if there exist $x_1,\cdots,x_n \in X$ such that $M=[d(x_i,x_j)]_{n\times n}$.
Question. For which ...
- 1,991
13
votes
8
answers
3k
views
Applications of the notion of of Gromov-Hausdorff distance
I am looking for applications of the notion of Gromov-Hausdorff convergence to prove theorems that a priori have nothing to do with it. Examples that I am aware of (thanks to wikipedia and google):
...
58
votes
14
answers
19k
views
Open problems in Euclidean geometry?
What are some (research level) open problems in Euclidean geometry ?
(Edit: I ask just out of curiosity, to understand how -and if- nowadays this is not a "dead" field yet)
I should clarify a bit ...
1
vote
1
answer
908
views
What are the topological properties of the metric space retained (inherited) for its completion
Let $(X,d)$ be a metric space and $(\bar{X},\bar{d})$ its completion.
There is a list of topological properties
Wikipedia - Topological property
Does anybody know list which of them are retained (...
31
votes
11
answers
16k
views
Is there any geometry where the triangle inequality fails?
We know from elementary school that the triangle inequality holds in Euclidean geometry. Some where in High School or in Univ., we come across non-Euclidean geometries (hyperbolic and Riemannian) and ...
- 2,855
2
votes
0
answers
254
views
Forgetting extra structure inducing Symmetries
This is a major edit of the original post after receiving helpful comments.
It is often the case when one adds additional structure to make a problem more tractable. When one attempts to forget this ...
124
votes
37
answers
12k
views
One-step problems in geometry
I'm collecting advanced exercises in geometry. Ideally, each exercise should be solved by one trick and this trick should be useful elsewhere (say it gives an essential idea in some theory).
If you ...