11
votes
Isometric embedding of a genus g surface
The Nash-Kuiper Theorem implies the answer is yes if you only require the embedding to be of class $C^1$. However, I believe the actual visualization problem for $g\geq 2$ (that is, producing the ...
- 8,529
9
votes
Isomorphic embedding of $l^n_{\infty}$ into $l_1^m$?
No. Best embedding constant of $\ell^k_{\infty}$ into $L_1$ is of the order $\sqrt{k}$. This follows from the facts that $L_1$ has cotype 2 while the cotype 2 constant of $\ell^k_{\infty}$ is $\sqrt{k}...
- 2,429
8
votes
Accepted
Trigonometry / Euclidean Geometry for natural numbers?
It can be done for the metric
$$d(a,b)^2 = 1 - \frac{(a,b)}{\sqrt{ab}},$$
and other similar ones like $d(a,b)^2 = 1 - \frac{(a,b)^2}{ab}$, with some twists in the construction.
Suppose we want ...
- 1,235
8
votes
Accepted
Isometric embeddings of metric $K_{n+1}$ in $\mathbb{R}^n$
With the $\ell_\infty$-norm this is true. For example, it is a classic theorem of Fréchet that every $n$-point metric space embeds in $\ell_\infty^{n-1}$. The required embedding $f$ is easy to define. ...
- 32.1k
7
votes
Accepted
Bilipschitz embedding of the unit ball of $c_0$ into $\ell_1$
Yes, this is known. Raynaud showed that $B_{c_0}$ does not uniformly (in particular, bilipschitz) embed into any stable Banach space. $\ell_1$ is stable.
Yves Raynaud, Espaces de Banach superstables, ...
- 2,429
7
votes
Accepted
When is a metric space a snowflake?
You may be interested in the following paper:
Jeremy T. Tyson and Jang-Mei Wu, Characterizations of Snowflake Metric Spaces. Annales Academiae Scientiarum Fennicae Mathematica. Volume 30, 2005, 313-...
- 86
7
votes
Accepted
Are hyperbolic spaces actually better for embedding trees than Euclidean spaces?
I am not sure if the following paper answers your question. The abstract suggests so, but it is written in a computer science style that is less transparent to me in terms of stating a precise theorem....
- 86
7
votes
Accepted
Explicit formula for embedding Cayley graph of free group into hyperbolic space
The subgroup $\Gamma < \mathrm{SL}(2, \mathbb{R})$ generated by the matrices
$
a =
\begin{pmatrix}
1 & 2 \\
0 & 1
\end{pmatrix}
$
and
$
b =
\begin{pmatrix}
1 & 0 \\
2 & 1
\end{...
- 28.2k
7
votes
Isometric embedding of a genus g surface
I don't know the answer to your question, and I wouldn't be surprised if it were open. You may be interested to know that any compact Riemannian two-manifold $(V, g)$ admits a $C^{\infty}$ isometric ...
- 19.3k
6
votes
Canonical immersion of the double torus
This is not really an answer, but rather a longish comment and a suggestion about how one might focus the question a bit better.
First, when one asks for a 'canonical' isometric embedding into some ...
- 108k
5
votes
Accepted
Does there exist a countable metric space which is Lipschitz universal for all countable metric spaces?
The affirmative answer to this problem follows from
Lemma. For any countable dense subsets $X,Y$ in the half-line $\mathbb R_+=[0,+\infty)$ there exists a $C^2$-smooth function $f:\mathbb R_+\to\...
- 41.9k
5
votes
Rozendorn's Article
If you don't read Russian: Rozendorn's construction is presented by Aminov in Extrinsic geometric properties of the Rozendorn surface, which is an isometric immersion of the Lobachevskiĭ plane into $E^...
- 39.1k
5
votes
Accepted
Is $\ell_p$ $(1<p<\infty)$ finitely isometrically distortable?
We (me and the author posing the question) have answered this question in the negative - for each $p \neq 2$ we have found a space $X$ such that $X$ is isomorphic to $\ell_p$ but there are 5 element ...
- 538
4
votes
Trigonometry / Euclidean Geometry for natural numbers?
Given that the set of integers has fractal dimension -1, I would not be surprized that such trigonometry is possible, it would be trigonometry on a manifold of negative dimension. Particularly, the ...
- 10.1k
4
votes
Accepted
Fast Bourgain embedding (or similar embeddings)?
There is a way to speed-up Bourgain's embedding in case if the original metric space has low "intrinsic dimension". The resulting algorithm will have a theoretical runtime $O(CN\log^2(N))$, where $C$ ...
- 1,058
4
votes
Volume of submanifold as integral of delta-function
One should distinguish between the volume of the submanifold (a number that might be infinite) and the volume form, an exterior differential form $\omega$ of degree $n{-}m$ on the (presumed regular) ...
- 108k
4
votes
Accepted
Bi-Lipschitz embeddings of compact doubling spaces
No. There are many compact, doubling metric spaces with no bi-Lipschitz embedding in a Hilbert space. Some examples:
The closed unit ball in the (continuous) Heisenberg group with its Carnot-Carath'...
- 56
3
votes
Accepted
Rozendorn's Article
If you don't mind looking at the original Russian version, then the scanned article itself is available at
https://arar.sci.am//Content/23775/file_0.pdf
I found it by searching https://www.google.com/...
- 346
3
votes
Are hyperbolic spaces actually better for embedding trees than Euclidean spaces?
Here is a trivial example for question B (in the $s=1$ case):
The discrete metric space on $N$ points with distance 1 between every two distinct points has the minimal distortion of an embedding into $...
- 148k
3
votes
Does every CAT(0) space embed in a measurable integral of $\mathbb{R}$-trees?
It meant to be an answer to another question, https://mathoverflow.net/a/406833/, but it is an answer to this question as well.
- 45k
3
votes
Accepted
Embedding Turing machine
Let $\mathcal{M}$ be a class of binary functions acting on strings in $\Sigma^*$,
along with a "size" function $|\cdot|:\mathcal{M}\to\{1,2,\ldots\}$ with the property that there are only finitely ...
- 6,541
3
votes
Accepted
Banach embedding of finite dimensional spaces
For $2<r<\infty$, if $\ell_s^n$ embeds uniformly into $\ell_r$ for all $n$, then either $s=r$ or $s=2$. This is basically the localization to finite dimensions of the classical dichotomy theorem ...
- 31.5k
3
votes
Embedding a countably infinite metric space in $\ell^2(\mathbb Z_+)$
When embedding $f_n:\{ x_1, \ldots , x_{n+1}\}\to\mathbb R^n\cong \ell^2(\{ 1, \ldots ,n\})$, we can assume that $f_n(x_j)\in \ell^2(\{ 1, \ldots , j-1\})$ (and $f_n(x_1)=0$), by giving $\ell^2$ a ...
- 24.2k
3
votes
Accepted
Is the face lattice of the cube a polytope graph?
Yes. The Hasse diagram of any polytope is the edge graph of the dual of its antiprism.
- 2,773
3
votes
Is the face lattice of the cube a polytope graph?
Let me expand on Daniel's answer. The paper Daniel cites defines the antiprism (p.4) of an abstract polytope $\mathcal P$ to be
$$\operatorname{Ant}(\mathcal P):=\{(F,G)\mid F,G\in\mathcal P \text{ ...
- 13.6k
2
votes
Accepted
Lower Estimate of A Lipschitz Map
Of course, there might be such a function $\rho$ for a specific $f$, but there need not be one in general.
Let $X$ be $\mathbb{R}^2$, $Y$ be $\mathbb{R}$ and let $f(x,y)$ be $x$ if $x<0$ and $0$ ...
- 96
2
votes
Accepted
Suppose that a metric space allows David–Semmes regular map to some Euclidean space. Does it allow bi-Lipschitz embedding into some Euclidean space?
In some sense, this is probably the most unsatisfactory answer possible, but it's all I know:
In the 2005 paper 1, Movahedi-Lankarani and Wells write the following in VII (p. 16):
Tomi Laakso (6) ...
- 61
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