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Metric currents on singular measures in $\mathbb R^d$

Unless I am misunderstanding a lot of works, it is my understanding that a finite and non negative measure $\mu=g\mathcal{H}^\alpha$, where $\mathcal{H}^\alpha$ is the $\alpha$-Haudorff measure, ...
Lolman's user avatar
  • 391
0 votes
1 answer
114 views

Geometric interpretation of a Grammian-like function

Let $\mathbf{v}, \mathbf{w} \in \mathbb{R}^n$ and consider the following function $f : \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$: $$ f(\mathbf{v},\mathbf{w}) = \|\mathbf{v}\|\|\mathbf{w}...
Nathaniel Johnston's user avatar
5 votes
1 answer
483 views

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
2 answers
349 views

Mutual metric projection

Given a subset $S\subseteq \mathbb{R}^n$, the metric projection associated with $S$ is a function that maps each point $x\in \mathbb{R}^n$ to the set of nearest elements in $S$, that is $p_S(x) = \arg ...
Erel Segal-Halevi's user avatar
0 votes
0 answers
119 views

Boundedness of 2 times the unit ball

Suppose that $X$ is a topological vector space where the topology is given by a metric $d$ on $X$. Assuming that the unit ball $$ B(0, 1) := \{x \in X : d(0, x) < 1\} \neq X, $$ is it necessarily ...
Chandan Biswas's user avatar
9 votes
2 answers
471 views

Proving the inequality involving Hausdorff distance and Wasserstein infinity distance

Prove the inequality $$d_{H}(\mathrm{spt}(\mu),\mathrm{spt}(\nu))\leq W_{\infty}(\mu,\nu)$$ where $d_H$ denotes the Hausdorff distance between the supports of the measures $\mu$ and $\nu$, and $W_\...
Luna Belle's user avatar
1 vote
0 answers
128 views

Sum of upper semi continuous and lower semi continuous functions

Let $X$ be a compact metric space. Assume that $f: X \to \mathbb{R}$ is upper-semi continuous and $g:X \to \mathbb{R}$ is lower semi-continuous. Assume that $\sup \{ f(x)+g(x) : x \in X \}$ is finite. ...
Adam's user avatar
  • 1,043
3 votes
1 answer
161 views

Equivalent definition for Skorokhod metric

I have a question about the Skorokod distance on the space $\mathcal{D}([0,1],\mathbb{R})$: $$ d(X,Y):= \inf_{\lambda \in \Lambda}\left( \sup_{t\in [0,1]}|t-\lambda(t)|\vee \sup_{t\in [0,1]}|X(t)-Y(\...
user1598's user avatar
  • 177
3 votes
0 answers
87 views

Instances of c-concavity outside of optimal transport?

Let $X$ and $Y$ be metric spaces, and let $c:X\times Y\rightarrow \mathbb{R}$ be a nonnegative function which we refer to as a cost. For any $\phi:X\rightarrow \mathbb{R}$ and $\psi:Y\rightarrow \...
Brendan Mallery's user avatar
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
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
2 votes
0 answers
78 views

Converse of existence of minimizers

Let $(V,\|\cdot\|)$ be a real normed linear space. $V$ has the property that given any nonempty convex, closed subset $K$, there exists a unique $v_0\in K$ such that $\|v_0\| \leq \|v\|, \forall v\in ...
Rohan Didmishe's user avatar
1 vote
0 answers
146 views

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^...
ABIM's user avatar
  • 5,405
7 votes
2 answers
434 views

Vector measures as metric currents

Currents in metric spaces were introduced by Ambrosio and Kirchheim in 2000 as a generalization of currents in euclidean spaces. Very roughly, a principle idea is to replace smooth test functions (and ...
Jochen Wengenroth's user avatar
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. ...
user982564'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
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 ...
Markus's user avatar
  • 1,361
28 votes
6 answers
12k views

Almost orthogonal vectors

This is to do with high dimensional geometry, which I'm always useless with. Suppose we have some large integer $n$ and some small $\epsilon>0$. Working in the unit sphere of $\mathbb R^n$ or $\...
Matthew Daws's user avatar
  • 18.7k
40 votes
5 answers
5k views

"Entropy" proof of Brunn-Minkowski Inequality?

I read in an information theory textbook the Brunn-Minkowski inequality follows from the Entropy Power inequality. The first one says that if $A,B$ are convex polygons in $\mathbb{R}^d$, then $$ m(...
john mangual's user avatar
  • 22.8k
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
1 answer
110 views

Lipschitz maps with Hölder inverse preserve the doubling property

Let $K$ be a compact doubling metric space, $X$ be a metric space and $f:K\rightarrow X$ be Lipschitz with $\alpha$-Hölder inverse, where $0<\alpha<1$. Does $f(K)$ need to be doubling?
ABIM's user avatar
  • 5,405
4 votes
0 answers
132 views

A Lipschitzian's condition for the measure of nonconvexity

I'm actually working on the measure of nonconvexity and its application. Especially, the Eisenfeld–Lakshmikantham MNC defined - in a Banach space - by: $$\alpha(A)=\sup_{b\in\overline{\operatorname{...
Motaka's user avatar
  • 291
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
0 votes
0 answers
56 views

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’...
Brendan Mallery's user avatar
13 votes
0 answers
818 views

Covering number estimates for Hölder balls

Let $\alpha \in (0,1]$, $r>0$ and $L>0$, and positive intwgers $n$ and $m$. The Arzela-Ascoli Theorem guarantees that the set $X(\alpha,L,r)$ of $f:[-1,1]^n\rightarrow [-r,r]^m$ with $\alpha$-...
ABIM's user avatar
  • 5,405
6 votes
1 answer
260 views

Arbitrary-dimensional expanders?

Rephrasing expansion (slightly). Consider the following slightly tweaked version of the usual definition of a (spectral) expander graph. (We write a weighted graph as $(V,\beta)$, where the weight $\...
H A Helfgott's user avatar
  • 20.2k
8 votes
1 answer
4k views

Covering number of Lipschitz functions

What do we know about the covering number of $L$-Lipschitz functions mapping say, $\mathbb{R}^n \rightarrow \mathbb{R}$ for some $L >0$? Only 2 results I have found so far are, That the $\infty$-...
gradstudent's user avatar
  • 2,246
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
5 votes
2 answers
245 views

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&...
AngeloPiadetta's user avatar
2 votes
0 answers
92 views

Linearization stability condition

The following is a theorem from Fischer and Marsden's 1975's paper: Linearization stability of nonlinear PDEs. Theorem. Let $X, Y$ be Banach manifolds and $\Phi: X \rightarrow Y$ be $C^1$. Let $x_0 \...
Gordhob Brain's user avatar
4 votes
2 answers
353 views

Does $C[0, 1]$ admit a covering by sets of arbitrarily small eccentricity?

We denote by $C[0, 1]$ the space of continuous functions on $[0, 1]$ under the supremum norm, equipped with the Borel sigma algebra. A covering of $C[0, 1]$ is a (possibly countably infinite) ...
Nate River's user avatar
  • 6,215
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
12 votes
1 answer
1k views

Smoothness of distance function to a compact set

Fix a non-empty compact subset $K\subseteq \mathbb{R}^n$ and let $d_K(x):=\min_{z \in K} \,\|z-x\|$ be the map sending any $x\in \mathbb{R}^n$ to its distance from $K$. Suppose that: $K$ is regular : ...
ABIM's user avatar
  • 5,405
1 vote
1 answer
174 views

Dimension-preserving non-linear map

Let $F:\mathbb{R}^n\to\mathbb{R}^n$ be a continuous non-linear map, and let $A$ be a connected subset of $\mathbb{R}^n$ with $\text{dim}(A)=d\leq n$. When can we say that the dimension of the image, $\...
RS-Coop's user avatar
  • 39
13 votes
3 answers
3k views

Are uniformly continuous functions dense in all continuous functions?

Suppose that $X$ is a metric space. Is the family of all real-valued uniformly continuous functions on $X$ dense in the space of all continuous functions with respect to the topology of uniform ...
user124775's user avatar
3 votes
1 answer
350 views

Talagrand's inequality for L1 norm

I have a series of $n$ independent random variables $X_1,\ldots, X_n$, each with the support $[0,1]$, and a monotone convex function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ that is 1-Lipshitz in L1 ...
Tomer Ezra's user avatar
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 $(\...
Yury Korolev's user avatar
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 ...
Tomasz Kania's user avatar
  • 11.3k
2 votes
1 answer
169 views

Metric analogue of upper/lower semicontinuity

Let's say we have two metric spaces, $(X,\rho)$ and $(Y,\tau)$. The continuity of $f:X\to Y$ is obvious and natural to define. What about semi-continuity? Without a natural ordering on $Y$, perhaps &...
Aryeh Kontorovich's user avatar
2 votes
2 answers
261 views

Distribution of the support function of convex bodies: beyond mean width

Let $K$ be a symmetric convex body in $\mathbb{R}^n$ (that is the unit ball of a norm). Let $h_K$ be its support function, that is $h_K(u) = \sup_{x \in K}\langle x,u \rangle$. The quantity $w(K) = \...
Gericault's user avatar
  • 245
2 votes
0 answers
99 views

Anisotropic Calderon-Zygmund decomposition

I am looking for the following version of Calderon-Zygmund decomposition, consider an function $f \in L^1(R^{d+1})$ and cylinders of the form $Q_{R,R^p}$ for some fixed $p \in (0,\infty)$, The ...
Adi's user avatar
  • 455
1 vote
0 answers
75 views

$L^1$-valued Lipschitz extension problem on a simplex

Consider a regular $n$-simplex, and a map from the vertices to $L^1$. How can we find the minimum Lipschitz constant of an extension of this map to the entire simplex? Is there any literature or ...
alesia's user avatar
  • 2,772
2 votes
0 answers
57 views

Does the snowflake $X^\alpha$ allows isometric embeddings into $L_1$ if $X$ does?

Question: Suppose that finite metric space $X$ allows isometric embedding to $L_1$. Does it mean that a snowflake space $X^\alpha$ allows isometric embedding to $L_1$ for every $0 < \alpha < 1$? ...
Vladimir Zolotov's user avatar
6 votes
0 answers
182 views

Factorization of metric space-valued maps through vector-valued Sobolev spaces

Let $(X,d,m)$ and $(Y,\rho,n)$ be metric measure spaces and let $f:X\rightarrow Y$ be a Borel-measurable function for which there is some $y_0$ and some $p\geq 0$ such that $$ \int_{x\in X}\,d(y_0,f(x)...
ABIM's user avatar
  • 5,405
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
1 vote
0 answers
449 views

Bound on covering number of Lipschitz functions – missing part in proofs of Kolmogorov et al

Given a metric space $(\mathcal{X},\rho)$ and $\mathcal{A}\subset\mathcal{X}$ totally bounded, i.e. $\mathcal{A}$ has a finite $\varepsilon$-covering for any $\varepsilon>0$. Consider $\...
samuel's user avatar
  • 11
5 votes
0 answers
74 views

Concentration bound on additive functions with constraints

Given a family of sets $F \subseteq P(\{1,\ldots,n\})$. I define the function $f_F:[0,1]^n \rightarrow R$ to be $f_F(x_1,\ldots,x_n)= \max_{S \in F} \sum_{j \in S} x_j$. Given a series of independent ...
Tomer Ezra's user avatar
11 votes
7 answers
1k views

What are some interesting ways of making new metrics out of old metrics?

If $d(x,y)$ and $e(x,y)$ are metrics then $d(x,y)+e(x,y)$ and $\frac{d(x,y)}{1+d(x,y)}$ are metrics. If $d_i(x,y)$ for $i=1,\dots,n$ are metrics then so is $\sqrt{\sum_{i=1}^n{d_i^2(x,y)}}$ Are ...
Kim Greene's user avatar
  • 3,613
2 votes
0 answers
93 views

Finite approximations to the Kuratowski/Fréchet embedding

Let $(X,d)$ be a compact doubling metric space with doubling constant $C>0$. Let $\{\mathbb{X}_n\}_{n=0}^{\infty}$ be a sequences of finite subsets of $X$ with $$ \left\{B\left(x_k,\frac1{n}\right)...
Carlos_Petterson's user avatar
1 vote
1 answer
519 views

Asymptotic cone

Let $S$ be a subset in a real vector space $\mathbb{R}^n$. Define the asymptotic cone $S\infty:=\{y\in\mathbb{R}^n\mid\textrm{there exists a sequence }(y_k,\varepsilon_k)\in S\times\mathbb{R}^+\textrm{...
Hebe's user avatar
  • 951