All Questions
Tagged with fa.functional-analysis mg.metric-geometry
70 questions with no upvoted or accepted answers
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
2
votes
0
answers
127
views
Functional inequality under mean curvature flow
Let $\Sigma$ be a hypersurface in $\mathbb R^n$ and $\Sigma_t$ be a variation of $\Sigma$ under the mean curvature flow under an extra condition that ${\rm vol}_{n-1}(\Sigma)={\rm vol}_{n-1}(\Sigma_t)$...
- 143
2
votes
0
answers
319
views
Banach-Mazur distance from finite-dimensional subspaces of $\ell_p$ to the Hilbert space
I am reading a paper http://www.math.tamu.edu/~johnson/TF3.4.pdf by Bill Johnson and Andrzej Szankowski and having trouble grasping why
$d_n(Z_m) \leq d_n(\ell_{p_{m+1}} ) = n^{|p_{m+1}-2|}$ in the ...
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
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. ...
- 1,043
1
vote
0
answers
145
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^...
- 5,405
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 ...
- 1,361
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 ...
- 2,772
1
vote
0
answers
448
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 $\...
- 11
1
vote
0
answers
70
views
Injectivity of post-composition operator
Let $X$, $Y_1,Y_2$, and $Z$ be separable metric spaces. Let $C(X,Y)$ be the topological space of continuous functions from $X$ to $Y$ equipped with its compact-open topologies. Fix a continuous ...
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 ...
- 93
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{...
- 1,087
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 ...
- 11
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 ...
- 5,405
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,
$$
is it necessarily ...
0
votes
0
answers
56
views
Zero flux along lines
I am considering the $L^1$ ball in $\mathbb{R}^d$, and a conservative vector field $V$ on it, which arises as the gradient of a bounded, almost-everywhere Lipchitz-function. Denoting by $e_i$ as the i’...
- 232
0
votes
0
answers
165
views
Compact embedding of Lipschitz continuous functions
Let $(X,d,\mu)$ be a metric measure space, not necessarily with $\mu(X)<\infty$. I would like to study the embedding of $W^{1,2}(X)\cap \mathrm{Lip}(X)$ into $L^2(X)$. Are there simple conditions ...
- 3,388
0
votes
0
answers
44
views
Let $A,B,C$ be centrally-symmetric convex bodies. What is this function $G(x,y) := \sup_{b \in B}\inf_{a \in A} a^T x - b^T y + \|a-b\|_C$?
Let $A$, $B$, and $C$ be centrally-symmeric convex bodies in $\mathbb R^n$. Note that any such set can such set induces a norm $\|\cdot\|_C$ on $\mathbb R^n$ defined by $\|x\|_C := \sup_{c \in C}c^\...
- 6,853
0
votes
0
answers
254
views
The set of all functions which vanish at infinity is a subset of the set of all functions which have vanishing variation
Let $X$ be a coarse space, we define the following:
$D_b(X)$ is the set of all bounded functions $f:X\rightarrow \mathbb{C}$
$f\in $$D_b(X)$ is said to vanish at infinity if for each $\varepsilon$>0 ...
- 313
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 ...
- 1,095