All Questions
Tagged with embeddings metric-embeddings
16 questions
8
votes
1
answer
412
views
Embedding Euclidean buildings into products of trees
A Euclidean building has a natural metric space structure. (A definition of Euclidean building can be found on Wikipedia, or, more expansively, in Section 4 of Kleiner-Leeb.)
Question: Is it true ...
- 83
7
votes
1
answer
590
views
When is a metric space a snowflake?
Let $(X,d)$ be a metric space. For any $0<\epsilon<1$, we call the metric space $(X,d^{\epsilon})$; where $d^{\epsilon}(x,y)\triangleq (d(x,y))^{\epsilon}$ the $\epsilon$-snowflake of $(X,d)$.
...
5
votes
0
answers
178
views
Correspondence between Riemannian metrics and Euclidean embeddings
Given a sufficiently smooth manifold M,
a Riemannian metric on M induces an isometric embedding into Euclidean space by Nash's theorem, (non-canonically, non-uniquely)
an embedding of M into ...
4
votes
1
answer
700
views
Embedding graphs into hyperbolic spaces
Do we know of a characterization as to when does a graph have a "good" embedding into a hyperbolic space? (And does having such an embedding have a spectral or wavelet analysis signature?)
I don't ...
- 617
4
votes
0
answers
227
views
To what extent is the Nash embedding not unique?
Consider a smooth Nash embedding, $f$, of a Riemannian manifold $Σ$ into Euclidean space $\mathbb R^n$. To what extent is this embedding not unique?
It is clear that the set of all such embeddings ...
- 521
3
votes
1
answer
138
views
Banach embedding of finite dimensional spaces
Recall that: let $0<r<s<2$, then $\ell_r$ uniformly contains a subspace isomorphic to $\ell_s^m$, $m\ge 1$ (see [JS]).
I am wondering whether are any result for the case when $r>s>2$?
...
- 617
3
votes
0
answers
208
views
Volume of sub-manifold of $\mathbf R^n$
Let $M$ be an $n-m$ dimensional sub-manifold of $\mathbf R^n$ defined by the following set of polynomial equations:
\begin{equation}
P_1(\vec x)=0, \\
\vdots \\
P_m(\vec x)=0,
\end{equation}
(where $\...
- 521
3
votes
0
answers
158
views
Constant in Naor and Neiman's Assouad Theorem
In Naor and Neiman's Assouad embedding theorem - "Assouad’s theorem with dimension independent of the snowflaking" Revisita Mathematica, the authors derive quantitative estimates on the ...
- 5,405
2
votes
1
answer
353
views
Volume of submanifold as integral of delta-function
Let $M$ be an $n-m$ dimensional sub-manifold of $\mathbf R^n$ defined by the following set of equations:
\begin{equation}
f_1(\vec x)=0, \\
\vdots \\
f_m(\vec x)=0,
\end{equation}
(where $\vec x$ are ...
- 521
2
votes
0
answers
93
views
Finite approximations to the Kuratowski/Fréchet embedding
Let $(X,d)$ be a compact doubling metric space with doubling constant $C>0$. Let $\{\mathbb{X}_n\}_{n=0}^{\infty}$ be a sequences of finite subsets of $X$ with
$$
\left\{B\left(x_k,\frac1{n}\right)...
- 195
2
votes
0
answers
94
views
Almost Lipschitz embedding of compact metric measure spaces into Euclidean spaces
Let $(X,d)$ be a compact metric space, $m$ be a metric outer-measure on $X$. Are there 'mild conditions' on $X$ ensuring the existence of a positive integer $N\geq 3$ such that there exist $x_1,\dots,...
- 109
2
votes
0
answers
77
views
Dense embeddings into Euclidean space
The question is a follow-up on this old post. Fix a positive integer $d$ and consider $\mathbb{R}^d$ with its usual Euclidean topology. Given a metric space $(X,\delta_X)$, what conditions are ...
- 5,405
2
votes
0
answers
131
views
Embedding a binary subspace to $l_2$ in a much lower dimension
I'm trying to find a way to embed a binary linear subspace of dimension $n$ (a linear code) to the Euclidian space while reducing the dimension significantly.
The subspace (or code) contains points ...
- 301
1
vote
0
answers
125
views
Do cycle graphs embed isometrically in spheres?
I recently came across, what seems to be a folklore. Namely, that cycle graphs embeds isometrically into spheres $S^n(r)$, for some $n\in \mathbb{N}_+$ and some $r>0$. However, I could not track ...
- 169
1
vote
0
answers
106
views
Best estimate on doubling constant of a finite metric space
Let $(X,d)$ be a finite metric space. Clearly, $(X,d)$ is a doubling metric space but is there a 'best' estimate of $(X,d)$'s doubling constant?
Probability based on its cardinality, diameter, and ...
- 195
0
votes
1
answer
83
views
Lower Estimate of A Lipschitz Map
Suppose that $(X,d_X)$ and $(Y,d_Y)$ are complete doubling metric spaces and let $f:X\rightarrow Y$ be a non-constant Lipschitz map. Then can does there exist a lsc function
$\rho:(0,\infty)\...
- 5,405