All Questions
Tagged with metric-spaces gt.geometric-topology
9 questions
10
votes
1
answer
561
views
Does a compact contractible metric space have a point that is fixed by all isometries?
Let $(X,d)$ be a compact and contractible metric space. Let $\operatorname{Isom}(X)=\{\phi\colon X\to X\}$ be its group of isometries.
Question: Is there a point $x\in X$ fixed by all $\phi\in\...
- 13.6k
8
votes
2
answers
689
views
Topological characterization of injective metric spaces
Let $\ (X\ d)\ \,(Y\ \delta)\ $ be arbitrary metric spaces. A function $\ f:X\rightarrow Y\ $ is called a metric map (with respect to the given metrics $\ d\ \delta$) $\ \Leftarrow:\Rightarrow\ \...
- 7,500
6
votes
0
answers
813
views
Limit of metric spaces
Let $\{X_n\}_{n\in \mathbb{N}}$ be a collection of T2 topological spaces, with maps $f_n\colon X_n \to X_{n+1}$. These maps are continuous and open. Let $X$ be the direct limit of this system.
Assume ...
- 2,384
3
votes
1
answer
112
views
Independence of the axiomatics of metric cones
A metric cone $C$ is a nonempty metric space (whose distance is denoted $d$) together with a map $\cdot\colon \mathbf{R}\times C \mapsto C$ satisfying these axioms:
$a\cdot(b\cdot x) = (ab)\cdot x$ ...
- 33
3
votes
1
answer
244
views
Partitioning a smooth manifold into geodesically convex sets
Let $X$ be a connected and compact $d$-dimensional smooth manifold; where $d$ is a positive integer. Does (or rather, when does) there exist a metric $\rho$ on $X$ generating $X$'s topology and a ...
- 5,405
3
votes
0
answers
177
views
When do Polish spaces admit complete metric making them $\mathrm{CAT}(\kappa)$?
Question
$\DeclareMathOperator\CAT{CAT}$Let $X$ be a Polish space. When are there known conditions under which $X$'s topology can be metrized by a metric $d$ such that $(X,d)$ is a:
$\CAT(\kappa)$ ...
- 195
1
vote
0
answers
97
views
Are Hölder functions between Banach spaces residual in the compact-open topology?
Let $X$ and $Y$ be Banach spaces and let $C(X,Y)$ be the set of continuous functions from $X$ to $Y$ equipped with the topology of uniform convergence on compact sets (i.e. the compact-open topology). ...
- 5,405
0
votes
1
answer
514
views
Distance between two points using triangulation
Suppose we have two points $p_1$ and $p_2$ in a metric space with unknown dimensionality, with no way to directly compute the distance between them, e.g. no coordinates.
Say we can randomly sample a ...
0
votes
0
answers
169
views
Do all manifolds admit metrics with Euclidean balls?
Let $M$ be a compact topological n-manifold. Suppose we are given a locally flat embedding $M \subset \mathbb{R}^{n+k}$. This induces a metric on $M$ by restriction. Is it true that for $\epsilon$ ...
- 5,849