3
votes
1answer
165 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 ...
3
votes
0answers
85 views

construction of heat kernels for non-compact manifolds with boundary

Recently, I am studying heat semigroup for noncompact manifolds with boundary. In Issac Chavel's book "eigenvalues in Riemannian geometry". "Given a noncompact Riemannian manifold, it need not be ...
5
votes
3answers
227 views

When does heat kernel have both Gaussian upper and lower bounds?

Recently, I am reading Sturm's paper "Analysis on local Dirichlet form III: X is a locally compact separable Hausdorff space and m is a positive Radon measure with supp[m]=X. $\varepsilon$ is a ...
1
vote
1answer
124 views

heat kernel $p_t(x_0,y) \in D(\Delta) \cap L^\infty$ for a manifold with Ricci curvature bounded below?

X is an n-dim Riemannian manifold with the Dirichlet form $$ \varepsilon (u,v) =-\int_X \langle \nabla u,\nabla v \rangle $$ for $u,v \in W^{1,2}(X)$. Let $P_t$ and $p_t(x,y)$ be the associate ...
4
votes
1answer
77 views

generator of Dirichlet form coincide with the absolute part of the “Laplacian”

Let M be an Riemannian manifold with the Dirichlet form $$\varepsilon (u,v) =-\int_M \langle \nabla u,\nabla v \rangle$$ for $u,v \in W^{1,2}_0(M)$. Let $\Delta^M:D(\Delta^M) \to L^2(M)$ denote the ...
1
vote
1answer
61 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? $W^{1,2}(M)$ ...
2
votes
2answers
868 views

Heat flow $P_tf \to f$ in $W^{1,2}$ for $f \in W^{1,2}$?

$\varepsilon:L^2(X,m) \to [0,\infty]$ is a strongly local, symmetric Dirichlet form generating a Markov semigroup $(P_t)_{t\ge0}$ in $L^2(X,m)$. Let $D(\varepsilon)=\{f\in ...
6
votes
1answer
223 views

Does a metric refine the weak-* topology on a dual space?

Let $X$ be a topological affine space over $\mathbb C$, with no additional assumptions. Let $X^*$ denote its dual space of continuous affine functionals $X \to \mathbb C$, equipped with the weak-$*$ ...
7
votes
2answers
196 views

Distortion of tree embedding in Alexandrov spaces

It is a well-known theorem first proved by Bourgain that any map $\varphi:T_n\to H$ from the binary tree of height $n$ to a Hilbert space has distortion at least $C \sqrt{\ln n}$ where $C$ is a ...
7
votes
1answer
438 views

complete metric space

Hallo, I have the following question: Let $(X,d)$ be a complete metric space. Is then $(X,\operatorname{dist})$ also complete? Here by $\operatorname{dist}$ I mean the metric induced by $d$ by: ...
0
votes
1answer
172 views

The pth power of a distance function is twice continuously differentiable, for $p>2$?

Suppose $\mathcal{O}$ is an open convex connected strict subset in $\mathbb{R}^n$ and define $\beta(x)=dist(x, \mathcal{O})$, for each $x\in\mathbb{R}^n$. Is $\beta^p$, $p>2$ a twice continuously ...
5
votes
2answers
561 views

A characterization of Hilbert spaces?

My question was prompted by an earlier MO by @Daniel:     Duality map in strictly convex Banach spaces I will even use his symbol   $\phi$   below. Let   $B$   be an ...
5
votes
2answers
387 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 ...
3
votes
1answer
99 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 ...
5
votes
1answer
234 views

For which metric measure spaces is the Hardy-Littlewood maximal operator not of weak type (1,1)?

Let $(X,d,\mu)$ be a metric measure space, i.e. $\mu$ is a Borel measure on the metric space $(X,d)$. I'll denote the Hardy-Littlewood maximal operator - either centred or uncentred, I don't mind ...
2
votes
4answers
343 views

Continuous pointwise ergodic theorem?

Let $\Phi$ be a homeomorphism of a compact metric space $M$ which preserves a regular Borel probability measure $\mu$.(`Regular' $\mu(U) > 0$, if U open. ) Under these hypothesis, I have two ...
1
vote
1answer
71 views

Log-nonexpansive functions: terminology and references

During my recent work in the optimization of positive valued functions, the following class of functions proved to be exceptionally important. (Defn.). Let $h: (0,\infty) \to (0,\infty)$ be ...
5
votes
0answers
94 views

Regularity of simplices, part deux

This question is directly inspired by Pietro Majer's question and my answer to it. One can define a simplex, and the dihedral angles thereof in an infinite dimensional Hilbert space (one has to take ...
3
votes
1answer
166 views

Are faces of a compact, convex body “opposed” iff their extreme points are pairwise “opposed”?

Let $P$ be a compact, convex subset of $\mathbb{R}^n$ (infinite-dimensional generalisations welcome, but not necessary). Let's say that disjoint subsets $W_1$, $W_2$ $\subset P$ are opposed if there ...
0
votes
1answer
133 views

Does homeomorphism preserves the family of cones?

Let me state my problem. Suppose we have a ball $B$ in standard $\mathbb{R}^3$, that is a $\varepsilon$-neighbourhood of $0$ point. Suppose we have a family of cones $X_C = \lbrace C > 0 \vert x^2 ...
9
votes
1answer
504 views

Question on Hilbert Manifolds

I have a very basic question on Hilbert manifolds. Consider the Hilbert space $$ \mathcal{H}:= L^2(S^1) $$ with $S^1$ the unit circle. On $\mathcal{H}$ let us introduce the equivalence relation $$ ...
8
votes
0answers
371 views

High-dimensional geometry: Top-down Vs. Bottom-up

There are several ways to leverage one's intuition from low-dimensional geometry to understand high-dimensional phenomena. For example, one can get a clearer picture of the behaviour of ...
2
votes
0answers
139 views

Parameterizing infinitesimal perturbations of the sphere using signed measures

Let $\mathcal K^3$ be the space of convex bodies, with some metric $\delta$, and let $B$ be the unit ball. Let us define a volume-preserving perturbation of the sphere to be a continuous (substitute ...
5
votes
0answers
252 views

Local minimum from directional derivatives in the space of convex bodies

I have a function $f(K)$ defined on the space of three-dimensional convex bodies for which I want to show that the unit ball $B$ is a local minimum. I have been able to show if $K$ is not homothetic ...
2
votes
1answer
169 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 ...
2
votes
1answer
408 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 ...
2
votes
1answer
164 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 ...
8
votes
2answers
337 views

An extension of Gaussian Isoperimetry

The Gaussian isoperimetric inequality (Tsirelson,Sudakov, Borell) states that among all sets of given Gaussian measure in the n-dimensional Euclidean space, half-spaces have the minimal Gaussian ...
2
votes
0answers
490 views

Controlling the Lipschitz norm of the limit of a sequence of functions

Consider the Fr├ęchet space $\Omega = C(\mathbb R^d)$ of real-valued continuous functions equipped with the seminorms $$\|f\|_D := \sup_{x,y \in D} \left\{ |f(x)|, \tfrac{|f(x)-f(y)|}{|x-y|} \right\}, ...
7
votes
1answer
327 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 ...
15
votes
5answers
1k views

Unexpected applications of Dvoretzky's theorem

Dvoretzky's theorem is a classic of convex geometry. Recently at a conference in quantum information I learned (from Patrick Hayden's talk) about a nontrivial application of the theorem to a problem ...
2
votes
4answers
184 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) := ...
9
votes
2answers
557 views

Small crown probabilities (and infinite dimensional margin assumption)

My question is: How do I find sharp upper bounds on $P(|q|\leq \epsilon)$ uniformly over a set of gaussian polynomes $q$ of degree two. Notations and definitions (to make the question rigorous) ...
18
votes
1answer
4k views

L1 distance between gaussian measures

L1 distance between gaussian measures: Definition Let $P_1$ and $P_0$ be two gaussian measures on $\mathbb{R}^p$ with respective "mean,Variance" $m_1,C_1$ and $m_0,C_0$ (I assume matrices have full ...
12
votes
5answers
2k views

Almost orthogonal vectors

This is to do with high dimensional geometry, which I'm always useless with. Suppose with have some large integer n and some small $\epsilon>0$. Working in the unit sphere of $\mathbb R^n$ or ...
4
votes
2answers
499 views

Quantitative questions about the size of a finite epsilon net

Let $X$ be a metric space, and let $U \subset X$ be any set. A finite set $N = N(\epsilon) \subset U$ is called a finite $\epsilon$-net of $U$ if every point of $U$ is at most a distance of ...
11
votes
3answers
746 views

To what extent is convexity a local property?

A polyhedron is the intersection of a finite collection of halfspaces. These halfspaces are not assumed to be linear, i.e. their bounding hyperplanes are not assumed to contain the origin. The ...
8
votes
1answer
324 views

Estimating flat norm distance from a planar disc

Let $D\subset\mathbb R^2\subset\mathbb R^n$ be a unit planar disc in $\mathbb R^n$. Let $S$ be an orientable two-dimensional surface in $\mathbb R^n$ such that $\partial S=\partial D$. Of course, we ...
5
votes
3answers
588 views

How can I embed an N-points metric space to a hypercube with low distortion?

I have a N-point metric space defined by the pairwise distance matrix. I want to encode these N points with binary strings, i.e. each point will be mapped to a vertex in a hypercube. The lengths of ...
9
votes
7answers
654 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 ...