Questions tagged [metric-embeddings]
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68
questions
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1
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Is the face lattice of the cube a polytope graph?
The face lattice of a
convex polytope $P\subset\Bbb R^d$ is the partially ordered set whose elements are the faces of $P$ ordered by inclusion. We can turn it into a graph by considering its Hasse ...
- 12.8k
1
vote
1
answer
104
views
Alexandrov's uniqueness theorem in Minkowski spacetime
Suppose $P$ is a convex polyhedron in $\mathbb{R}^{2,1}$.
Each face of $P$ comes with induced metric tensor,
if the face is space-like, then it is euclidean metric;
every time-like face is isometric ...
- 44.4k
4
votes
1
answer
84
views
Embedding a countably infinite metric space in $\ell^2(\mathbb Z_+)$
Suppose $(X,d)$ is a countably infinite set endowed with a metric $d$ that satisfies the following condition:
Every finite subset of $X$ with the induced metric is isometric to a subset of some ...
- 2,598
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votes
0
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64
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Can the sequence of complete graphs coarsely embed into Hilbert space?
Basically the title. If I have the metric space which is the disjoint union of the sequence of complete graphs, and the usual graph metric, has it been shown that the metric space can be coarsely ...
- 53
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Is there a bi-Hölder Weierstrass-type embedding of the circle into some Euclidean space?
We say that $\Phi\colon S^1\to \mathbb{R}^d$ is an $\alpha$-bi-Hölder embedding if there are constants $c_1,c_2>0$, such that
$$c_1\leq \frac{\|\Phi(x)-\Phi(y)\|}{d(x,y)^\alpha}\leq c_2,$$
where $d$...
5
votes
1
answer
140
views
Variants of the Bonk-Schramm embedding
Recently I heard about the following embedding theorem of Bonk and Schramm: every Gromov hyperbolic geodesic metric space with "bounded growth" is roughly similar to a convex subset of $\...
2
votes
2
answers
195
views
Isometric embeddings of metric $K_{n+1}$ in $\mathbb{R}^n$
Question:
is it always possible to embed a complete, symmetric and metric graph $G$ with $n+1$ vertices isometrically in $\mathbb{R}^n$?
I'm convinced it must be true, but can't remember having seen ...
- 12.8k
1
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0
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119
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Intuition behind right-inverse of map from Johnson-Lindenstrauss Lemma
The Johnson–Lindenstrauss lemma states that for every $n$-point subset $X$ of $\mathbb{R}^d$ and each $0<\varepsilon\le 1$, there is a linear map $f:\mathbb{R}^d\to\mathbb{R}^{O(\log(n)/\varepsilon^...
- 5,079
4
votes
1
answer
187
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Bi-Lipschitz embeddings of compact doubling spaces
Suppose that $(X,\rho)$ is a compact doubling metric space. Does there necessarily exist an $\epsilon>0$ and a maximal $\epsilon$-net $\{x_i\}_{i=1}^n\subseteq X$ such that the map
$$
\begin{...
- 5,079
1
vote
0
answers
121
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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 ...
- 159
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votes
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344
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Embedding a graph into Euclidean space
I want to find a map $v\mapsto \tilde v$ from the vertex set of a connected infinite graph $\Gamma$ to a Euclidean space that meets the following two conditions:
there is $\varepsilon>0$ such that ...
- 44.4k
1
vote
1
answer
275
views
Bilipschitz embedding of the unit ball of $c_0$ into $\ell_1$
This is a follow-up to this question of mine.
It is well-known that the Banach space $\ell_1$ does not contain any isomorphic copies of $c_0$. One can even go further and show that $\ell_1$ does not ...
- 1,151
2
votes
1
answer
302
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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 ...
- 423
3
votes
0
answers
197
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 $\...
- 423
5
votes
1
answer
307
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Isomorphic embedding of $l^n_{\infty}$ into $l_1^m$?
Given $n$, is there a $C(n)$-isomorphic embedding of $l^n_{\infty}$ into $l^m_1$ for sufficiently large $m$ and $C(n)<<\log(n)$?
For $n=2$ this can done with $m=2$. There are some results about $...
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