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Can you always extend an isometry of a subset of a Hilbert Space to the whole space?

I remember that I read somewhere that the following theorem is true: Let $A\subseteq H$ be a subset of a real Hilbert space $H$ and let $f : A \to A$ be a distance-preserving bijection, i.e. a ...
Cosine's user avatar
  • 609
6 votes
1 answer
500 views

A characterization of metric spaces, isometric to subspaces of Euclidean spaces

I am looking for the reference to the following (surely known) characterization of metric spaces that embed into $\mathbb R^n$: Theorem. Let $n$ be positive integer number. A metric space $X$ is ...
Taras Banakh's user avatar
  • 41.9k
7 votes
1 answer
331 views

A metric characterization of Hilbert spaces

In the Wikipedia paper on Hadamard spaces, it is written that every flat Hadamard space is isometric to a closed convex subset of a Hilbert space. Looking through references provided by this Wikipedia ...
Taras Banakh's user avatar
  • 41.9k
5 votes
1 answer
244 views

Dual Banach space $X^*$ complemented in $\mathrm{Lip}_0(X)$?

$\DeclareMathOperator\Lip{Lip}$Let $X$ be a real Banach space. The dual $X^*$ is a closed subspace of $\Lip_0(X)$. ($\Lip_0(X)$ denotes the space of real-valued Lipschitz functions $f:X\to\mathbb{R}$ ...
Miek Messerschmidt's user avatar
4 votes
0 answers
147 views

Continuous extension preserving modulus of continuity

Let $X$ be a (non-empty) compact subset of $D(0,M):=\left\{x\in \mathbb{R}^n:\, \|x\|\leq M\right\}$, and let $f:X\rightarrow Y$ be uniformly continuous; for some metric space $Y$. Are there any ...
Catologist_who_flies_on_Monday's user avatar
2 votes
1 answer
223 views

Is there a theory of partially-defined metric spaces?

Is there a theory of metric spaces in which the distance between a given pair of points need not be defined? I'm aware that there is a theory of partial metric spaces, but these deal with a different ...
gmvh's user avatar
  • 3,065
1 vote
0 answers
122 views

Metric transforms that preserve $\ell^1$ embeddability

Consider a function $f$ from reals to reals such that $f$, when applied to pairwise Manhattan distances between $n$ points, always results in a set of Manhattan distances. Work by Schoenberg and ...
Timothy Chu's user avatar
3 votes
0 answers
89 views

Reference request: Projection operators in metric spaces

Given a metric space $(X,d)$ and a subset $S\subset X$, the projection $P_S$ onto $S$ is well-defined as a set valued function. I am interested in learning more about properties of these projections ...
JohnA's user avatar
  • 710
4 votes
0 answers
144 views

A Pythagorian inequality characterization of inner-product spaces

Let $(X,\|\cdot\|)$ be a real normed space. For any points $A$ and $B$ in $X$, let $AB:=\|A-B\|$. Suppose that for any points $A$ and $B$ in $X$ and any straight line $\ell\subseteq X$ such that $B\...
Iosif Pinelis's user avatar
16 votes
2 answers
731 views

A reference to a characterization of metric spaces admitting an isometric embedding into a Hilbert space

I am looking for a reference to the bipartite version of the Schoenberg's criterion of embeddability into a Hilbert space. The Schoenberg criterion is formulated as Proposition 8.5(ii) of the book &...
Taras Banakh's user avatar
  • 41.9k
6 votes
2 answers
457 views

$L^{p}$ isoperimetric inequalities on the Hamming cube

Let $A \subset \{-1,1\}^{n}$ be a subset of the Hamming cube with cardinality $|A|=2^{n-1}$. Define $w_{A} : \{-1,1\}^{n} \to \mathbb{N}\cup \{0\}$ so that $w_{A}(x)$ to be number of boundary edges ...
Paata Ivanishvili's user avatar
7 votes
0 answers
187 views

distance distributions on a hypersphere?

Fix a real number $0\leq t\leq 1$ and an integer $n>1$. Let $\mathbb{S}^{n-1}\subset\mathbb{R}^n$ denote the unit hypersphere. Define $$d_N(n;t):=\max\sum_{i<j}\Vert P_i-P_j\Vert_2^t$$ where ...
T. Amdeberhan's user avatar
2 votes
0 answers
127 views

Functional inequality under mean curvature flow

Let $\Sigma$ be a hypersurface in $\mathbb R^n$ and $\Sigma_t$ be a variation of $\Sigma$ under the mean curvature flow under an extra condition that ${\rm vol}_{n-1}(\Sigma)={\rm vol}_{n-1}(\Sigma_t)$...
Math101's user avatar
  • 143
5 votes
1 answer
328 views

Is a space with p-norm a Finsler manifold?

Suppose $\mathbb{R}^n$ is equipped with the p-norm $\left\Vert x \right\Vert_p$. Let $x\in \mathbb{R}^n$ and let $y$ be in a neighborhood of $x$. The distance between $x$ and $y$ can be defined as $\...
Klock's user avatar
  • 51
7 votes
1 answer
469 views

Embedding of real trees into $\ell_1(\Gamma)$

It seems plausible that any real tree or ${\mathbb{R}}$-tree in the sense of the definition in https://en.wikipedia.org/wiki/Real_tree admits an isometric embedding into the Banach space $\ell_1(\...
Mikhail Ostrovskii's user avatar
3 votes
1 answer
191 views

Maximal $\pi/2$-separated subset of the sphere

A subset $A$ of a metric space is called $\varepsilon$-separated if $$dist(x,y)> \varepsilon \mbox{ for all } x\ne y\in A.$$ (Notice that the inequality in my definition is strict.) What is the ...
asv's user avatar
  • 21.8k
8 votes
0 answers
421 views

Approximate singular value decomposition in Banach spaces

I am interested in generalisations to Banach spaces of the following construction, which relates to the singular value decomposition of a finite-dimensional linear map. If $V$, $W$ are finite-...
Ian Morris's user avatar
  • 6,206
7 votes
2 answers
1k views

For what spaces is the Hardy-Littlewood maximal operator of strong type $(p,p)$ if and only if $p > p_0 > 1$?

(This is essentially a continuation of my previous question, here.) Let $(X,d,\mu)$ be a metric measure space, i.e. $\mu$ is a Borel measure on the metric space $(X,d)$. Further assume (though you ...
user avatar
2 votes
1 answer
230 views

Completing The Space Sections in a Vectorbundle

Hi there. Assume $(M,g)$ is a Riemanian manifold and $E\to M$ is a vector bundle with a bundle metric $\langle\cdot,\cdot\rangle$. We then have the pre-Hilbert space $H_0:=\Gamma_c^\infty(E)$ of ...
Robert Rauch's user avatar
2 votes
1 answer
452 views

What do we get from an euclidian affine structure ?

Imagine you investigate a set of objects $\mathcal{E}$, and you just realize this that $\mathcal{E}$ possesses an affine structure with respect to some real vector space $\mathcal{V}$ having a scalar ...
Adrien Hardy's user avatar
  • 2,135