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Results tagged with metric-spaces
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user 172802
A metric space is a pair $(X,d)$, where $X$ is a set and $d:X \times X \to \mathbb{R}$ satisfies the following conditions for all $x,y,z \in X$. (Symmetry) $d(x,y)=d(y,x)$. (Identity of Indiscernibles) $d(x,y)=0$ if and only if $x=y$. (Triangle Inequality) $d(x,y)+d(y,z) \geq d(x,z)$.
17
votes
Accepted
Is the topology generated by this weaker notion of a metric necessarily metrisable?
For a loose metric $d$ as above, we can consider the function
$$d_1(x,y):=\sup\{|d(x,z)-d(y,z)|;z\in X\}.$$
It is easy to verify that $d_1$ is a metric, and $d(x,y)\leq d_1(x,y)\leq\rho(d(x,y))$ for a …
- 10.6k
10
votes
Accepted
Relationship between doubling constant of a metric space and of a metric measure space
Apart from the obvious counterexample of the measure being $0$, if $(X,d,m)$ is doubling in the sense of metric measure spaces it will be doubling in the sense of metric spaces.
Consider a ball $B(x,r …
- 10.6k
4
votes
1
answer
281
views
Is every 1-Lipschitz homeomorphism $f:X\to X$ from a compact metric space to itself an isome...
I found a statement involving a homeomorphism $f:X\to X$ of a compact metric space $X$, with Lipshitz coefficient 1, i.e., a non-expansive map, and cannot think of an example where $f$ is not an isome …
- 10.6k
2
votes
0
answers
106
views
Definition of the category QMet of metric spaces and quasi-isometries
I am following Clara Löh's Geometric Group Theory. An Introduction, and in remark 5.1.12, she defines the category QMet whose objects are metric spaces and whose morphisms are quasi-isometric imbeddin …
- 10.6k
1
vote
Changing a metric to that 2 points have different distance
Yes, such a distance $d'$ exists.
We can suppose $d(x,X)=d(y,X)=k$ for some $k>0$. We can define a new distance $d'$ by $d'(a,b)=d(a,b)+|d(x,a)-d(x,b)|$. This easily implies $d'(x,X)=2k$, however for …
- 10.6k
1
vote
1
answer
114
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How to find the point at minimal average distance of a given measure
Given a compactly supported probability measure $m$ on $\mathbb{R}^n$, we can define its average distance to a point $x$ as $\int_\mathbb{R^n}d(x,y)dm(y)$. In this question I found that for a given me …
- 10.6k
1
vote
Accepted
Lipschitz maps with Hölder inverse preserve the doubling property
If I have understood the definitions correctly, $f(K)$ need not be doubling.
For example consider a map $f$ from $[0,1]$ to the Hilbert space $\mathbb{R}\times l^2$ defined in the following way. Let $ …
- 10.6k