All Questions
37 questions
0
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56
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Zero flux along lines
I am considering the $L^1$ ball in $\mathbb{R}^d$, and a conservative vector field $V$ on it, which arises as the gradient of a bounded, almost-everywhere Lipchitz-function. Denoting by $e_i$ as the i’...
- 232
1
vote
0
answers
126
views
Non-surjective isometries of $l_p$
It is well known that all surjective isometries of $l_p$ for $p\neq 2$ are the signed permutations of the unit vector basis $(e_n)$. Is there a characterization for the linear non-surjective ...
- 1,361
9
votes
0
answers
1k
views
Weak compactness in $\mathcal{F}(X)$
Let $(X,0)$ be a pointed metric space and let $\mathcal{F}(X)$ be the natural predual of ${\rm Lip}_0(X)$, the space of Lipschitz functions on $X$ that map $0$ to $0$; here $\mathcal{F}(X)$ is really ...
- 11.3k
5
votes
1
answer
205
views
Existence of a Gelfand triple involving the Arens–Eells space (aka Lipschitz free space)
$\DeclareMathOperator\Lip{Lip}\DeclareMathOperator\AE{AE}$Background
Gelfand triples. Let $\mathcal B$ be a Banach space, $\mathcal B^*$ its dual space, and $\mathcal H$ a Hilbert space. The triple $(\...
- 437
4
votes
1
answer
207
views
Reference for Chebyshev centers
Today, I came across the concept of Chebyshev center twice.
In particular, it is the key tool in the very elegant paper "A fixed point theorem for $L^1$ spaces" by Bader, Gelander and Monod.
...
- 71
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 $f:X\to\mathbb{R}$ ...
5
votes
2
answers
245
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Differentiability of the map $x\mapsto \delta_x$ in the Arens-Eells/Lipschitz-free space
$\DeclareMathOperator\AE{AE}\DeclareMathOperator\Lip{Lip}$Let $\AE(X)$ denote the Arens-Eells space on a Banach space $X$. Consider the map:
$$
\begin{aligned}
\delta: X & \rightarrow \AE(X)
\\
x&...
6
votes
0
answers
132
views
Mazur-Ulam bases in finite-dimensional Banach spaces
Definition. A basis $e_1,\dots,e_n$ of a finite-dimensional Banach space $X$ is called Mazur-Ulam if all vectors $e_1,\dots,e_n$ have norm one and every self-isometry $f:S_X\to S_X$ of the unit sphere ...
- 4,535
2
votes
0
answers
159
views
Explicit homeomorphism between $L^p$ and Sobolev Space
From the Anderson-Kadec theorem, we know that all separable infinite-dimensional Banach spaces are homeomorphic. I'm wondering, is there an explicit such homeomorphism between $W^{p,k}(\mathbb{R}^n)$ ...
- 5,405
4
votes
1
answer
137
views
Are unit balls in Banach spaces retracts of bidual balls?
Let $X$ be a separable Banach space embedded canonically in $X^{**}$. Is there a retraction from the unit ball $B_{X^{**}}$ of $X^{**}$ onto the unit ball $B_X$ of $X$?
When we insist on uniformly ...
- 97
5
votes
0
answers
139
views
Copies of $\ell_\infty^k$ in subspaces of the space of operators between $n$-dimensional Banach spaces
Are there a positive integer $k$ and an unbounded increasing function $d:\mathbb N\to\mathbb N$ (of growth order $\Omega(n^2)$) such that for any $n$-dimensional Banach spaces $X,Y$, the Banach space $...
- 4,535
1
vote
1
answer
194
views
Strictly increasing functions in reflexive subspaces of $C([0,1])$
By the Banach-Mazur theorem, every separable Banach space $X$ embeds into $C([0,1])$. When $X$ is reflexive, it is not possible to find a sequence of disjointly supported, non-negative functions in ...
- 97
5
votes
1
answer
177
views
An extremal property of points on the unit sphere of a 2-dimensional Banach space
Let $(X,\|\cdot\|)$ be a 2-dimensional real Banach space and $S=\{x\in X:\|x\|=1\}$ be its unit sphere. Assume that $S$ is smooth in the sense that for any $y\in S$ there exists a unique functional $y^...
- 41.8k
1
vote
1
answer
896
views
Known Lipschitz-free spaces
The Lipschitz-Free space (also known as Arens-Eells spaces) $\mathcal{F}(X,d)$ over a pointed metric space $(X,d)$ is a well-studied object. In many instances, we have "concrete" representations of ...
- 5,405
14
votes
1
answer
922
views
What are the applications of the Mazur-Ulam Theorem?
Every bijective isometry between normed spaces is affine. This well-known and beautiful statement, the Mazur-Ulam Theorem, was proved in 1932, but the proof has been simplified and polished in years, ...
- 60.5k
3
votes
0
answers
103
views
"Hoelder conjugate" version of the Johnson-Lindenstrauss transform
A variation of the well-known Johnson-Lindenstrauss transform (JLT) asserts that for $x_1,\ldots,x_m\in\mathbb{R}^n$ there exists a linear transformation $A:\mathbb{R}^n\to\mathbb{R}^k$ with $k=\...
- 111
5
votes
0
answers
350
views
How to calculate the volume of a parallelepiped in a normed space?
Let $E$ be a real normed space, and let $v_1,...,v_n\in E$ be linearly independent. The parallelepiped defined by these vectors is $P=\{\sum_{i=1}^{n}\alpha_i v_i|~0\le\alpha_i\le 1\}$. Since $E$ is a ...
- 5,529
3
votes
1
answer
273
views
Predual to Lipschitz maps with $p$ derivatives
Let $p\in \mathbb{N}$, and define $\mathrm{Lip}_p$ be the collection of functions from $\mathbb{R}^d$ to itself, with $p-1$ first derivatives bounded and whose $p^{\mathrm{th}}$ derivative is ...
- 5,405
3
votes
2
answers
253
views
Reference request: $\alpha$-Hölder spaces as double duals
If $(X,d)$ is a complete metric space, we define the $\alpha$-Hölder class $\Lambda_\alpha(X)$ as the subset of $C_b(X)$ satisfying that
$$
\sup_{x \neq y} \frac{|f(x) - f(y)|}{|x - y|^\alpha}.
$$
...
- 1,090
5
votes
0
answers
231
views
Which subspaces of $\ell_p^n$ are isometric?
This question is similar to the one asked here:
Extending linear isometries from subspaces of $\ell_p^n$
Let $p$ be an even integer. Let $X,Y$ be subspaces of $\ell_p^n$, and let $U : X \to Y$ be a ...
- 53
6
votes
1
answer
450
views
Is each compact metric space a subset of a compact absolute 1-Lipschitz retract?
A metric space $X$ is called an absolute $L$-Lipschitz retract if for any metric space $Y$ containing $X$ there exists a Lipschitz retraction $r:Y\to X$ with Lipschitz constant $Lip(r)\le L$.
...
- 41.8k
3
votes
0
answers
109
views
Is the Banach space $C(K)$ a $1$-Lipschitz comp-extensor?
Given a real number $c\ge 1$ let us say that a metric space $X$ is a $c$-Lipschitz comp-extensor if each Lipschitz self-map $f:K\to K$ of a compact subset $K\subset X$ extends to a Lipschitz self-map $...
- 41.8k
8
votes
1
answer
360
views
What is the smallest Lipschitz constant of a Lipschitz retraction of $\ell_\infty([0,1])$ onto $C[0,1]$?
By Theorem 1.6 in the book "Geometric Nonlinear Functional Analysis" by Benyamini and Lindenstrauss, the Banach space $C[0,1]$ is a Lipschitz retract of the Banach space $\ell_\infty[0,1]$. ...
- 41.8k
3
votes
1
answer
232
views
Extending linear isometries from subspaces of $\ell_p^n$
Take $p\in (1,\infty)\setminus \{2\}$. Let $X$ be a subspace of $\ell_p^n$ and let $U\colon X\to \ell_p^m$ ($m\geqslant n$) be a linear isometry. Is it possible to extend $U$ to a (non-surjective) ...
- 331
4
votes
1
answer
193
views
A bound on the square distance of a random walk on undirected graph
Fact:
Let $G$ be an $n$-vertex undirected graph and $(X_s)_{s\in \mathbb N}$ a stationary random walk on $G$. Then for every $s\in \mathbb{N}$,
$ \mathbb{E}[d_G(X_s,X_0)^2] \le C s \log n $, for some ...
- 401
2
votes
1
answer
167
views
Distortion of embedding in Hilbert space
Given an injective linear map $T$ between Banach spaces $X$ and $Y$, let
\begin{equation} d(T) = \sup \left \{ \frac{||x||_X}{||Tx||_Y}: x \in X \mbox{ is nonzero } \right\} \cdot ||T||_{\mathrm{op}}...
- 1,769
14
votes
0
answers
205
views
Have there been further developments on this scheme for polytope approximations to the unit ball of $\ell_p^n$?
A long time ago I happened to look at, and save (on a floppy disk!) for future reading, a copy of the following article:
W. T. Gowers, Polytope approximations of the unit ball of $l^n_p$.
In Convex ...
- 25.8k
2
votes
0
answers
319
views
Banach-Mazur distance from finite-dimensional subspaces of $\ell_p$ to the Hilbert space
I am reading a paper http://www.math.tamu.edu/~johnson/TF3.4.pdf by Bill Johnson and Andrzej Szankowski and having trouble grasping why
$d_n(Z_m) \leq d_n(\ell_{p_{m+1}} ) = n^{|p_{m+1}-2|}$ in the ...
1
vote
2
answers
165
views
Antiproximanal subspace of $L_1[0,1]$
Could someone give a reference or construct an example of closed subspace of $Y\subset L_1[0,1]$ such that $\operatorname{dist}(x,Y)$ is not attained of for any $x\notin Y$.
I read somewhere that $Y$ ...
- 1,697
12
votes
1
answer
575
views
Is $\ell_p$ $(1<p<\infty)$ finitely isometrically distortable?
Let $Y$ be a Banach space isomorphic to $\ell_p$, $1<p<\infty$. Is it true that any finite subset of $\ell_p$ is isometric to some finite subset of $Y$?
It seems to me that it is an interesting ...
- 4,878
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(\...
- 4,878
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-...
- 6,206
17
votes
0
answers
488
views
Large almost equilateral sets in finite-dimensional Banach spaces
Question: Does there exist a function $C:~(0,1)\to
(0,\infty)$ such that for each $\varepsilon\in(0,1)$ every Banach space
$X$ of dimension $\ge C(\varepsilon)\log n$ contains an $n$-point
set $\{x_i\...
- 4,878
4
votes
1
answer
438
views
Characterization of $l_p$ up to a linear isometry
There is a science called "Geometry of Banach spaces". I wonder if they managed to give a geometric characterization of $\ell_p$ ($p\in[1,\infty]$) up to isometric isomorphism (among all Banach spaces)...
- 7,426
5
votes
1
answer
808
views
Separable Banach spaces which are absolute Lipschitz retracts
A subset $F$ of a metric space $M$ is called a Lipschitz retract of $M$ if there is a Lipschitz map from $M$ onto $F$ which coincides with the identity on $F$. A metric space which is a Lipschitz ...
- 607
6
votes
1
answer
249
views
What is the doubling dimension of convex functions?
I am interested in the complexity of convex functions, specifically the "doubling dimension" of the class of convex functions defined on a compact subset of Euclidean space, when compared using the $L^...
- 173
7
votes
1
answer
362
views
Nonexpansive multi-valued maps in $\ell^2$
Let $C$ be a nonempty bounded closed convex subset, say the unit ball, of $\ell^2(\mathbb{N})$. Let $T: C\to 2^C$ be a map such that $T(x)$ is nonempty closed for each $x$, and that $$D(Tx,Ty)\le \|x-...
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