All Questions
189 questions
4
votes
1
answer
262
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 ...
- 95
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{...
- 363
6
votes
3
answers
1k
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 ...
- 171
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 ...
- 2,480
8
votes
2
answers
289
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 ...
- 14.4k
4
votes
1
answer
439
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
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
1
vote
2
answers
909
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 L^2(X,m):\varepsilon(f)<\...
- 199
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
1
vote
1
answer
75
views
Name for a uniform local boundedness property of a function
I am working with a function $f : \mathbb{R}^N \to \mathbb{R}$ having the property that for every $R > 0$, there exists $M > 0$ such that if $x, y \in \mathbb{R}^N$ and $\vert x - y \vert \le R$,...
- 2,834
4
votes
0
answers
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
$$ \|X\|_{p_1,p_2} = \left( \sum_{j=1}^n \left( \sum_{i=1}^m |x_{ij}|...
- 980
12
votes
3
answers
530
views
Making an l_2 distance out of l_1 distance
If we think of the l1 distance as a grid-distance between points, then we can think of l2 distance as what we get when we "shortcut" the grid by going "inside" a cell.
Making the grid finer doesn't ...
- 4,462
6
votes
1
answer
591
views
For which metric measure spaces is the Hardy-Littlewood maximal operator not of weak type (1,1)?
Let $(X,d,\mu)$ be a metric measure space, i.e. $\mu$ is a Borel measure on the metric space $(X,d)$. I'll denote the Hardy-Littlewood maximal operator - either centred or uncentred, I don't mind ...
user14166
6
votes
2
answers
1k
views
Quantitative questions about the size of a finite epsilon net
Let $X$ be a metric space, and let $U \subset X$ be any set. A finite set $N = N(\epsilon) \subset U$ is called a finite $\epsilon$-net of $U$ if every point of $U$ is at most a distance of $\epsilon$...
- 943
0
votes
1
answer
277
views
both convex and superharmonic function on manifold concave?
M is a non-compact Rimannian manifold without boundary.
$f\in W_{loc}^{1,2}(M)$ satisfies $\Delta f \leq c$ in the weak sense, i.e.
$$
-\int_M \langle \nabla f,\nabla \psi \rangle dvol \leq c\int_M \...
- 689
9
votes
1
answer
544
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
$$
...
- 233
6
votes
1
answer
396
views
Does a metric refine the weak-* topology on a dual space?
Let $X$ be a topological affine space over $\mathbb C$, with no additional assumptions. Let $X^*$ denote its dual space of continuous affine functionals $X \to \mathbb C$, equipped with the weak-$*$ ...
- 8,512
6
votes
1
answer
249
views
What is the doubling dimension of convex functions?
I am interested in the complexity of convex functions, specifically the "doubling dimension" of the class of convex functions defined on a compact subset of Euclidean space, when compared using the $L^...
- 173
5
votes
0
answers
394
views
construction of heat kernels for non-compact manifolds with boundary
Recently, I am studying heat semigroup for noncompact manifolds with boundary.
In Issac Chavel's book "eigenvalues in Riemannian geometry". "Given a noncompact Riemannian manifold, it need not be ...
- 199
2
votes
1
answer
452
views
What do we get from an euclidian affine structure ?
Imagine you investigate a set of objects $\mathcal{E}$, and you just realize this that $\mathcal{E}$ possesses an affine structure with respect to some real vector space $\mathcal{V}$ having a scalar ...
- 2,135
1
vote
1
answer
196
views
heat kernel $p_t(x_0,y) \in D(\Delta) \cap L^\infty$ for a manifold with Ricci curvature bounded below?
X is an n-dim Riemannian manifold with the Dirichlet form
$$
\varepsilon (u,v) =-\int_X \langle \nabla u,\nabla v \rangle
$$
for $u,v \in W^{1,2}(X)$.
Let $P_t$ and $p_t(x,y)$ be the associate ...
- 199
3
votes
0
answers
189
views
Can we obtain topology results using analysis in metric measures spaces?
Let $M$ be a smooth compact manifold. It is known that a lower bound on the Ricci curvature is equivalent to the convexity of the entropy on $\mathcal{P}^2(M)$ (Von Rennesse and Sturm '05), but I don'...
- 215
9
votes
2
answers
477
views
An extension of Gaussian Isoperimetry
The Gaussian isoperimetric inequality (Tsirelson,Sudakov, Borell) states that among all sets of given Gaussian measure in the n-dimensional Euclidean space, half-spaces have the minimal Gaussian ...
- 949
0
votes
1
answer
208
views
The pth power of a distance function is twice continuously differentiable, for $p>2$?
Suppose $\mathcal{O}$ is an open convex connected strict subset in $\mathbb{R}^n$ and define $\beta(x)=dist(x, \mathcal{O})$, for each $x\in\mathbb{R}^n$.
Is $\beta^p$, $p>2$ a twice continuously ...
- 9
4
votes
1
answer
251
views
generator of Dirichlet form coincide with the absolute part of the "Laplacian"
Let M be an Riemannian manifold with the Dirichlet form $$\varepsilon (u,v) =-\int_M \langle \nabla u,\nabla v \rangle$$ for $u,v \in W^{1,2}_0(M)$. Let $\Delta^M:D(\Delta^M) \to L^2(M)$ denote the ...
- 199
9
votes
2
answers
674
views
Small crown probabilities (and infinite dimensional margin assumption)
My question is:
How do I find sharp upper bounds on $P(|q|\leq \epsilon)$ uniformly over a set of gaussian polynomes $q$ of degree two.
Notations and definitions (to make the question rigorous)
Let ...
11
votes
0
answers
601
views
High-dimensional geometry: Top-down Vs. Bottom-up
There are several ways to leverage one's intuition from low-dimensional geometry to understand high-dimensional phenomena. For example, one can get a clearer picture of the behaviour of high-...
- 1,666
2
votes
0
answers
800
views
Controlling the Lipschitz norm of the limit of a sequence of functions
Consider the Fréchet space $\Omega = C(\mathbb R^d)$ of real-valued continuous functions equipped with the seminorms $$\|f\|_D := \sup_{x,y \in D} \left\{ |f(x)|, \tfrac{|f(x)-f(y)|}{|x-y|} \right\}, \...
- 8,512
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
$$ d(A, B) := \...
- 21
1
vote
1
answer
304
views
How do maximum norms relatively change in Euclidean translations
Let $Q$ be the cube $[-1,1]^{3}$ and $\pi$ be a plane in $\mathbb{R}^{3}$
that contains the origin but doesn't contain any vertex of $Q$. Suppose that $A$ is an invertible
linear transformation from $\...
- 11
2
votes
1
answer
134
views
Dirichlet energy with domain $W^{1,2}(M)$ or $W^{1,2}_{loc}(M)$ can be a specific Dirichlet form?
M is a Riemannian manifold, $\varepsilon(f,g)=\int_M \langle {\nabla f,\nabla g}\rangle dvol$.
Then with which domain is $\varepsilon$ a strongly local, regular and tight Dirichlet form?
$W^{1,2}(M)$ ...
- 199
6
votes
0
answers
387
views
Local minimum from directional derivatives in the space of convex bodies
I have a function $f(K)$ defined on the space of three-dimensional convex bodies for which I want to show that the unit ball $B$ is a local minimum. I have been able to show if $K$ is not homothetic ...
- 5,971
8
votes
1
answer
381
views
Estimating flat norm distance from a planar disc
Let $D\subset\mathbb R^2\subset\mathbb R^n$ be a unit planar disc in $\mathbb R^n$. Let $S$ be an orientable two-dimensional surface in $\mathbb R^n$ such that $\partial S=\partial D$. Of course, we ...
- 32.4k
7
votes
1
answer
362
views
Nonexpansive multi-valued maps in $\ell^2$
Let $C$ be a nonempty bounded closed convex subset, say the unit ball, of $\ell^2(\mathbb{N})$. Let $T: C\to 2^C$ be a map such that $T(x)$ is nonempty closed for each $x$, and that $$D(Tx,Ty)\le \|x-...
- 744
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 ...
- 23
0
votes
1
answer
156
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 +...
- 165
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 ...
1
vote
1
answer
111
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 ...
- 28.6k
5
votes
0
answers
104
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 ...
- 96.4k