All Questions
Tagged with metric-spaces isometry
199
questions
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Can a finite Wasserstein metric on Euclidean support be embedded in a Euclidean space?
Thanks for everyone's help with understanding finite metric embeddings in Euclidean space. I have a follow-up question.
Say we have the Wasserstein distance between $n$ distributions in Euclidean ...
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1
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55
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Conditions on a finite metric that guarantees embedding in Euclidean space? [duplicate]
If we have $n$ points in some metric space, do there exist coordinates for the $n$ points in an $n-1$ dimensional Euclidean space with exactly the same pairwise distances as in the original space?
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74
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The isometry group of a metric space with the topology of pointwise convergence is a topological group
Let $(X,d)$ be a metric space and let $\mathrm{Iso}(X)$ denote the group of isometries of $(X,d)$, where the group operation is the composition. Equip $\mathrm{Iso}(X)$ with the topology of pointwise ...
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When is isometry space of compact metric space a manifold?
Supposse we have a compact metric space $(X, d)$. Let $(\text{Iso}(X, d), D)$ be a metric space of all isometries $f : X \rightarrow X$ with a metric $D$ defined with:
$$D(f, g) = \text{sup}\{d(f(x), ...
- 271
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98
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If $A$ and $B$ are countable dense subsets of compact metric space $(X, d)$ with infinitely many isometries, is there isometry such that $f(a) \in B$.
Let $(X, d)$ be a separable and compact metric space with infinitely many isometires. Let $A, B$ be infinitely countable dense subsets of $X$. Is there an isometry $f : X \rightarrow X$ such that $f(a)...
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1
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73
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Is isometric to $\mathbb R^n$ implies vector space?
Suppose $(E,d_E)$ is a metric space that is isometric to $\mathbb R^3$ with Euclidean distance. I have some impedent :
(1) I want to state that $E$ is a metric space of dimension $n$, but as far as I ...
- 579
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2
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302
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Orbits of points in compact metric space when there exists a point that is fixed by all isometries
Let $(X,d)$ be a compact metric space and Iso$(X,d)$ be a set of all isometries $f : X \rightarrow X$.
For $x \in X$ orbit of $x$ is defined as: $$\operatorname{Orb}(x) := \{f(x) : f \in \operatorname{...
- 271
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1
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A sufficient condition for an isometry to exist between metric spaces
Reminder: An isometry of metric spaces $(X,d)$, $(Y,d')$ is a function
$$f:X \to Y$$
Such that for any $s,t \in X$ we have
$$d(s,t) = d(f(s),f(t))$$
i.e. It preserves all distances.
A homeomorphism of ...
- 3,044
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103
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Metric Space with No Local Isometries?
We know there are metric spaces with no non-trivial (global) isometries. For example, consider the Euclidean plane $(\mathbb{R}^2, d)$. Let $X \subset \mathbb{R}^2$ be the set $\{(0,0), (1,0),(0,2)\}$....
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prove that every isometry Is also continous [duplicate]
Let $ (X,d_x) $ and $(Y,d_y)$ be metric spaces, an isometry $f:(x,d_x)\rightarrow (y,d_y)$
Is Always continous
Proof:
A function Is an isometry if keeps distances unaltered therfore $ \forall x_1,x_2\...
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1
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213
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Show that $T$ is a topological embedding of $X$ in itself.
Let $X$ be the set of odd positive fractions with denominator $3$ in lowest terms, except for $\frac13$.
$X=\{x\in\Bbb Q^+:3x\in3\Bbb N\pm1\}\setminus\{\frac13\}$
If $a,b$ are 2-adic units, then any 2-...
- 9,279
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What is the sum of the 2-adic values of the orbits of the affine group over 2-adic integers restricted to $ax+b$ where $2^{\nu_2(x)}=2^{\nu_2(b)}$?
Let $\textrm{aff}(\Bbb Z_2)$ be the affine group over 2-adic integers, defined as the linear polynomials $ax+b$ where $a\in\Bbb Z_2^\times$ and $b\in\Bbb Z_2$.
A very interesting subset of this group ...
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1
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64
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Isometry Between C[0,1] and C[1,2]
This is not homework, I am just trying to make something clear for myself.
I am trying to show that $C([0,1])$ and $C([1,2])$ are isometric. Here both spaces are endowed with the standard sup metric. ...
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Construct on $\mathbb{R}^2$ a norm $\lVert \cdot \rVert$ such that the metric space ($\mathbb{R}^2 , \lVert \cdot \rVert$) is neither isometric to the
Construct on $\mathbb{R}^2$ a norm $\lVert \cdot \rVert$ such that the metric
space ($\mathbb{R}^2 , \lVert \cdot \rVert$) is neither isometric to the space ($\mathbb{R}^2 , \lVert \cdot \rVert_2$), ...
- 1,135
1
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1
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72
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If $f_n$, f are isometries, then $\underset{n\to\infty}{\lim}\overset{\infty}{\underset{i=0}{\sum}}2^{-i-1}d(f_n(x_i), f(x_i))=0$.
The Problem: Suppose $(X, d)$ is a complete separable metric space, and $\{x_i\}\subseteq X$ is a countable dense subset. Suppose $(f_n)$ is a sequence of isometries of $X$ such that $\underset{n\to\...
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