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4 votes
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244 views

On the modulus of convexity of mixed-norm $\ell_{p_1,p_2}$ spaces

Let $\ell_{p_1,p_2}=(\mathbb{R}^{m\times n},\|\cdot\|_{p_1,p_2})$ be the space of $m\times n$ matrices endowed with the mixed-norm $$ \|X\|_{p_1,p_2} = \left( \sum_{j=1}^n \left( \sum_{i=1}^m |x_{ij}|...
Cristóbal Guzmán's user avatar
3 votes
2 answers
675 views

distributional Hessian for semiconvex functions on non-smooth manifolds

Let $f:R^n \to R$ be convex. Then there exist signed Radon measures $\mu^{ij}=\mu^{ji}$ such that $$ \int_{R^n} f \frac{\partial^2 \varphi}{\partial x_i \partial x_j} dx= \int_{R^n} \varphi d\mu^{ij} ...
mafan's user avatar
  • 471
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
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
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
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
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
3 votes
1 answer
233 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) ...
user512365'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
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
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
3 votes
0 answers
239 views

How to prove the following the set are equal

Let $D$ be an open convex subset of $\mathbf R^d$ with boundary $\partial D$ and closure $\overline D$. For $x\in\mathbf R^d$ and a unit vector $e$ let $L_{x,e}=\{x+\lambda e, \lambda\ge0\}$ be the ...
Ailiy Evan's user avatar
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
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
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
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
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
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 $...
Taras Banakh's user avatar
  • 41.9k
3 votes
0 answers
108 views

Radial Poincare inequality for Gaussian measures

Let $\mu$ be a zero mean Gaussian probability measure on $\mathbb{R}^n$ whose covariance is less than the identity. If $f$ is a $1$-Lipschitz real function on $\mathbb{R}^n$ such that there exists a ...
alesia's user avatar
  • 2,772
3 votes
0 answers
82 views

Proving the existence of a continuous function that satisfy a certain property from a finite version of this property

Let $M \subseteq [0,1] \times \mathbb{R}^n$ be a compact semialgebraic set. In particular, $M$ can be described by a finite set of polynomial equalities and inequalities. Let $\delta_0 > 0$ be a ...
Eilon's user avatar
  • 745
3 votes
0 answers
189 views

Can we obtain topology results using analysis in metric measures spaces?

Let $M$ be a smooth compact manifold. It is known that a lower bound on the Ricci curvature is equivalent to the convexity of the entropy on $\mathcal{P}^2(M)$ (Von Rennesse and Sturm '05), but I don'...
Mario's user avatar
  • 215
2 votes
4 answers
222 views

How to compare finite point sets in normed spaces?

I want to define a "distance" between two subsets $A, B$ of a normed space $(V, \|\cdot\|)$ both with (at most) $n$ elements. A straightforward way for me to do this would be to define $$ d(A, B) := \...
Mirko's user avatar
  • 21
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
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
2 votes
1 answer
316 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
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
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
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
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
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}}...
burtonpeterj's user avatar
  • 1,769
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
208 views

Is there an elementary proof for preserving inequalities under the change of l_p metrics?

Here is what I mean exactly: Let $A=(a_1,a_2)$ and $B=(b_1,b_2)$ be two points in the real plane (for simplicity, but general finite dimensions would also be nice), and define the $\ell_p$-metric as ...
Fred von Heymann's user avatar
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
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
2 votes
1 answer
146 views

Prove a consequence of Poincare inequality and volume doubling

The question is Lemma 5.3 in [1] (with-out detailed proof). But I don't know how to prove. Let $M$ be a (finite dim) manifold satisfying the following two assumptions: (1) for any $x\in M$, and any ...
user84068's user avatar
  • 169
2 votes
1 answer
134 views

Dirichlet energy with domain $W^{1,2}(M)$ or $W^{1,2}_{loc}(M)$ can be a specific Dirichlet form?

M is a Riemannian manifold, $\varepsilon(f,g)=\int_M \langle {\nabla f,\nabla g}\rangle dvol$. Then with which domain is $\varepsilon$ a strongly local, regular and tight Dirichlet form? $W^{1,2}(M)$ ...
wang mu's user avatar
  • 199
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
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
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
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
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
2 votes
0 answers
71 views

Perturbing the approximation property from the Lipschitz-free space to stay in the Wasserstein space

Let $(X,d,x)$ be a separable pointed metric space and let $\mathcal{F}(X)$ be its Arens-Eells (also called its Lipschitz-Free space; in the case where $X$ is Banach) space. We view the $1$-...
ABIM's user avatar
  • 5,405
2 votes
0 answers
102 views

What is the relationship between barycenters in the Arens-Eells sense and barycenters in the optimal transport sense

Setup: Let $X$ be a complete pointed metric space. Let us briefly recall that the Wasserstein space $W_1(X)$ is identifiable with a subset of the Arens-Eells (or Lipschitz-Free) space $\operatorname{...
ABIM's user avatar
  • 5,405
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
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
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
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
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
2 votes
0 answers
60 views

Mean width of intersection of two elipsoid

My question is regarding mean widths. For a set $\mathcal{T}$ define the mean width \begin{align*} \omega(T)=\mathbb{E}_{\mathbf{g}\sim\mathcal{N}(0,\mathbf{I})}\bigg[\underset{\mathbf{u}\in\mathcal{...
Anahita's user avatar
  • 363
2 votes
0 answers
92 views

Estimating the size of a subset of $\mathbb{R}^N$

This concrete geometric question has arisen out of the problem of counting arithmetic functions with a particular property. The details of the relationship between the counting procedure and this ...
Kevin Smith's user avatar
  • 2,480