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2 votes
0 answers
186 views

Metric on space of Borel-measurable functions

Let $(X,d_X),(Y,d_Y)$ be metric spaces and $X$ is locally-compact and fix a Borel probability measure $\nu$ on $X$. For any Borel-measurable $f:X\rightarrow Y$, let $\mathcal{K}(f,\delta)$ be the set ...
Bernard_Karkanidis's user avatar
8 votes
1 answer
422 views

Why $(\mathrm{Lip}([0,1]^2))^*$ is finitely representable in 1-Wasserstein space over the plane?

In "Snowflake universality of Wasserstein spaces"" by Alexandr Andoni, Assaf Naor, and Ofer Neiman, they have the following notation: For a metric space X they write $\mathcal{P}_1(X)$ ...
Vladimir Zolotov's user avatar
2 votes
0 answers
49 views

A question about strong slopes (nonsmooth analysis)

Context. I'm reading the manuscrip "Nonlinear Error Bounds via a Change of Function" by Dominique Azé and Jean-Noël Corvellec (J Optim Theory Appl 2016), and I'm having a hard time ...
dohmatob's user avatar
  • 6,853
3 votes
1 answer
295 views

BV spaces and fractals -- are they Sobolev? Besov?

Do the real-valued functions of bounded variation on $[0,1]$ belong to some Sobolev/Besov class? What about a fractal, such as the Weierstrass function?
Aryeh Kontorovich's user avatar
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
1 vote
1 answer
195 views

Metric / strong slope restriction of function on unit ball in $\mathbb R^m$

Diclaimer. I'm not sure this is the right venue for this question, but I'll give it a try Definition [Strong / metric slope]. Given a complete metric space $(M,d)$ and a function $f:M \to (-\infty,+\...
dohmatob's user avatar
  • 6,853
2 votes
1 answer
173 views

Can we define geodesic in the space of compactly supported functions?

From Wikepedia, the definition of geodesic is stated as: A curve $\gamma: I\to M$ from an interval $I$ of the reals to the metric space $M$ is a geodesic if there is a constant $v\geq 0$ such that ...
mw19930312's user avatar
16 votes
1 answer
537 views

Balls in Hilbert space

I recently noticed an interesting fact which leads to a perhaps difficult question. If $n$ is a natural number, let $k_n$ be the smallest number $k$ such that an open ball of radius $k$ in a real ...
Bruce Blackadar's user avatar
2 votes
0 answers
265 views

The contraction principle in quasi metric spaces

I am researching contractive mappings and I need the article of I. A. Bakhtin "The contraction principle in quasi metric spaces"(1989) or at least part where explanation is given for ...
Dušan Bajović's user avatar
12 votes
5 answers
1k views

Examples of metric spaces with measurable midpoints

Given a (separable complete) metric space $X=(X,d)$, let us say $X$ has the measurable (resp. continuous) midpoint property if there exists a measurable (resp. continuous) mapping $m:X \times X \to X$ ...
dohmatob's user avatar
  • 6,853
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 $W^{p,k}(\mathbb{R}^n)$ ...
ABIM's user avatar
  • 5,405
4 votes
0 answers
97 views

Smoothing continuous functions in metric space

Let $(X,\rho)$ be a metric space. For any $f:X\to\mathbb{R}$, define the local Lipschitz constant of $f$ at $x$ by $$ \Lambda_f(x) := \sup_{x'\in X\setminus\{x\}} \frac{|f(x)-f(x')|}{\rho(x,x,')} . $$...
Aryeh Kontorovich's user avatar
1 vote
0 answers
97 views

Determining the behavior of a contraction mapping with undefined points

Label $X$ as the real interval $[0, a]$ where $a \in \mathbb{R}^+$, so that $\text{int}(X) = (0, a)$ labels the interior of $X$ and $\partial X$ labels the boundary of $X$. I have a function $f:\text{...
user918212's user avatar
  • 1,087
3 votes
0 answers
222 views

Sets of finite perimeter: intersection with an half space

I have a question regarding sets of finite perimeter. In particular I'm interested to find $$\mu_{E \cap H_t}, \label{1}\tag{1}$$ where $E$ is a set of finite perimeter in a generic open set $\Omega \...
ty88's user avatar
  • 51
3 votes
1 answer
309 views

Continuity/Lipschitz regularity of exponential map from $C_c$ to $\operatorname{Diff}_c$?

For finite-dimensional Lie algebras, see this for a nice example, the exponential map is smooth and in particular, it is locally-Lipschitz onto its image. However, things are different when moving to ...
ABIM's user avatar
  • 5,405
0 votes
0 answers
254 views

The set of all functions which vanish at infinity is a subset of the set of all functions which have vanishing variation

Let $X$ be a coarse space, we define the following: $D_b(X)$ is the set of all bounded functions $f:X\rightarrow \mathbb{C}$ $f\in $$D_b(X)$ is said to vanish at infinity if for each $\varepsilon$>0 ...
Hussain Rashed'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
1 answer
243 views

Choosing a convergent sub-sequence from a sequence of bi-Lipschitz homeomorphisms

Let $X,Y$ be two compact metric spaces. Suppose there is a sequence of bi-Lipschitz homeomorphisms $f_n: X\to Y$, and $c\in(0,1]$, satisfying $$c\cdot d(x_1,x_2)\le d(f(x_1),f(x_2))\le \frac{1}{c}\...
aglearner's user avatar
  • 14.3k
2 votes
1 answer
315 views

Estimate the metric entropy of unit ball in $L^2$ space

Let me clarify the setting I'm thinking. For any totally bounded metric space $(Y,d_Y)$ and $\varepsilon>0$, the $\textit{metric entropy}$ $N_M(\varepsilon,Y)$ is the smallest number of closed ...
S.Lim's user avatar
  • 469
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
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 ...
A. U.'s user avatar
  • 97
1 vote
0 answers
91 views

Gaussian width and restricted isometry

It is known that, for an $m$ dimensional space and an $n\times m$ dimensional random matrix $U$ whose entries are iid Gaussian, then $\|I-(1/n)U^TU\|$ is bounded by $\sqrt{m/n}$ when $n>m$. If a ...
Yue Sun's user avatar
  • 11
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 $...
Lviv Scottish Book's user avatar
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 ...
A. U.'s user avatar
  • 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^...
Taras Banakh's user avatar
  • 41.8k
0 votes
1 answer
223 views

Dense $G_{\delta}$ set with $\sigma$-porous complement is cofinite?

Let $X$ be a separable Banach space and $D\subseteq X$ be a proper, connected, and dense $G_{\delta}$ subset of $X$, $X-D$ is $\sigma$-porous. Then is $X-D$ contained in a finite-dimensional ...
ABIM's user avatar
  • 5,405
1 vote
0 answers
84 views

A Hölder version of the Johnson-Lindenstrauss Lemma on essentially bounded functions

Does there exist a Hölder (not necessarily linear) projection from $L^{\infty}(\mathbb{R}^d)$ to any finite-dimensional linear subspace? This is known when $L^{\infty}(\mathbb{R}^d)$ is replaced by a ...
ABIM's user avatar
  • 5,405
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 ...
ABIM's user avatar
  • 5,405
5 votes
2 answers
514 views

Concrete description of lift in Arens-Eells space

Let $X$ be a compact pointed metric subspace of the $d$-dimensional Euclidean space $(\mathbb{R}^d,d_E)$ and let $AE(X)$ denote its Arens-Eells space. Then a result of Nik Weaver shows that for every ...
ABIM's user avatar
  • 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, ...
Pietro Majer's user avatar
  • 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=\...
user134977's user avatar
2 votes
1 answer
145 views

Orbit-based metric

Let $(X,d)$ be a metric space and $f:X\rightarrow X$ be continuous. Then, is there any meaning/research done on the metric $$ D(x,y)\triangleq \sum_{n \in \mathbb{N}} \frac1{2^n} d(f^n(x),f^n(y)); $$ ...
ABIM's user avatar
  • 5,405
0 votes
0 answers
79 views

Hausdorff distance restricted to linear subspaces

Let $V$ be a Hilbert space, $Q \subset V$ be convex and compact and $Q_n \subset V$ be convex and compact for $n\in \mathbb{N}$ such that $Q_n \rightarrow Q$ for $n\rightarrow \infty$ in Hausdorff ...
Steve's user avatar
  • 1,095
2 votes
0 answers
90 views

Invariance under diffeomorphisms of the Hajlasz-Sobolev spaces

In this post it was shown that if $\Omega$ and $\Omega'$ are diffeomorphic non-empty open domains in some Euclidean space then the corresponding local Sobolev spaces are diffeomorphic with ...
ABIM's user avatar
  • 5,405
6 votes
1 answer
549 views

Volume doubling, uniform Poincaré, counterexample

The Poincaré inequality and the volume doubling property are important notions related to heat kernel estimates. Pavel Gyrya and Laurent Saloff-Coste obtain the two sided heat kernel estimate of ...
sharpe's user avatar
  • 721
6 votes
1 answer
348 views

Reference: Hajlasz-Sobolev Spaces with Values in a Metric Space

Let $(X,d,\mu)$ be a separable metric measure space on which every ball has positive but finite measure. I've come across the definition of a homogeneous Fractional Hajlasz-Sobolev spaces $M^{s,p}(...
ABIM's user avatar
  • 5,405
8 votes
2 answers
1k views

Talagrand's inequality for the discrete cube

Talagrand showed that if $f$ is a convex $1$-Lipschitz function on $\mathbb{R}^n$, and if $\mu$ is a product of probability measures supported over the interval, then $f$ has Gaussian concentration w....
alesia's user avatar
  • 2,772
3 votes
0 answers
487 views

Homeomorphism between $L^p$-spaces on metric spaces and $L^p$-spaces on Euclidean space

Setup: Fix $p \in [1,\infty)$. Let $(X,d_X,x_0)$ and $(Y,d_Y,y_0)$ be complete pointed metric spaces and $\mu$ be Borel. Let $E^n,E^D$ be Euclidean spaces of respetive dimensions $n$ and $D$ and ...
ABIM's user avatar
  • 5,405
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 ...
erz's user avatar
  • 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 ...
ABIM's user avatar
  • 5,405
2 votes
1 answer
210 views

$L^{2}$ Betti number

Let $\tilde{X}$ be a non-compact oriented, Riemannian manifold adimits a smooth metric $\tilde{g}$ on which a discrete group $\Gamma$ of orientation-preserving isometrics acts freely so that the ...
user94640's user avatar
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 $$ \sup_{x \neq y} \frac{|f(x) - f(y)|}{|x - y|^\alpha}. $$ ...
Adrián González Pérez's user avatar
3 votes
0 answers
86 views

What kind of set is this, spanned by two positive definite matrices?

Let $A$ and $B$ be Hermitian positive definite $n\times n$ matrices over $\mathbb C$ or $\mathbb R$. Then for real $k,\ell,$ the matrix $A^kB^\ell A^k$ is well-defined and again Hermitian positive ...
Wolfgang's user avatar
  • 13.4k
12 votes
2 answers
2k views

How to think about dual space of a certain space of Lipschitz functions

Consider the following Banach space (for concreteness): $$X=Lip(\bar{\mathbb{B}}^n)=\{f\in C^0(\bar{\mathbb{B}}^n): \Vert f \Vert_L<\infty \}$$ where $$ \bar{\mathbb{B}}^n=\{\mathbf{x}\in \mathbb{...
RBega2's user avatar
  • 2,478
6 votes
1 answer
896 views

Flat norm metrizes the weak* topology

I've come across the following statement in literature (without proof or reference) about the flat norm of currents $$ F(T) = \sup \{ T(\omega) : \omega \in D^k(U), |\omega(x)| \leq 1, |d\omega(x)| \...
golden-rabbit's user avatar
1 vote
1 answer
428 views

Growth rate of bounded Lipschitz functions on compact finite-dimensional space

Let $\mathcal X$ be a metric space of diameter $D$ and "dimension" (e.g doubling dimension) $d$. Let $L \in [0, \infty]$ and $M \in [0, \infty)$ and consider the class $\mathcal H_{M,L}$ of $L$-...
dohmatob's user avatar
  • 6,853
21 votes
1 answer
690 views

Diameter of a quotient of the infinite dimensional sphere

Suppose a group $\Gamma$ acts by isometries on the Hilbert space $\mathbb{H}^\infty$ and it fixes the origin. So $\Gamma$ acts on the unit sphere $\mathbb{S}^\infty$ as well. Assume that the action $...
Anton Petrunin's user avatar
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 ...
user127987's user avatar
7 votes
2 answers
664 views

Non-separable metric probability space

Let us say a metric probability space $(X,\rho,\mu)$ has property (*) if: the support of $\mu$ is contained in a separable subspace of $X$. Questions: 1. Is there a standard name for this property? ...
Aryeh Kontorovich's user avatar
13 votes
3 answers
650 views

General principles which lead to good questions in many concrete situations [closed]

I believe that in various fields of mathematics there are general principles which might lead to good questions and good results in many concrete situations. I would like to have a list of such ...