All Questions
11 questions
4
votes
1
answer
407
views
Lipschitz-regularity of partition of unity
Let $K$ be a compact subset of $\mathbb{R}^n$ and $\mathcal{U}$ be a finite collection of open subsets covering $K$ satisfying the minimality property: for every $U\in \mathcal{U}$, the sub-collection ...
- 195
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(\...
- 177
2
votes
2
answers
163
views
Monotonicity of doubling dimension
Let $(X,d)$ be a metric space with finite Assouad dimension $0<C_X$. It seems intuitive to me that if $\emptyset \subset Y\subseteq X$ then $Y$ is also doubling and its Assouad dimension, denoted ...
2
votes
1
answer
96
views
Isometry between punctured sphere and punctured triangle?
Setup:
Let $C_n$ be a closed $n$-simplex in $\mathbb{R}^n$ and let $r \in (0,R)$ where $R$ is the distance any one of the vertices $\{v_1,\cdots , v_{n+1}\}$ of $C_n$ to the centroid $\frac{v_1+ \...
- 5,405
1
vote
1
answer
158
views
Effect of snowflaking on doubling constants
This question is related to this one. Let $(X,d)$ be a metric space, let $\epsilon\in [0,1)$ and consider the snowflake $(X,d^{1-\epsilon})$. Suppose that $(X,d)$ has a finite doubling constant, ...
- 5,405
1
vote
1
answer
279
views
Inferring the modulus of continuity
Let $f:X\rightarrow Y$, $g:Y\rightarrow Z$ be uniformly continuous functions between metric spaces $X,Y,Z$ with moduli of continuity $\omega_f$ and $\omega_g$, respectively. Suppose that we know that ...
- 5,405
1
vote
0
answers
165
views
Uniformly open map on a dense subset
Schauder's lemma asserts that you can always extend a uniformly continuous, uniformly open map from a dense subset of a complete metric space to a uniformly open map on the completion.
I think the ...
- 11
0
votes
1
answer
228
views
Uniform distance from a discontinuous function is continuous
Define the metric $d(f,g)\triangleq \sup_{x \in [0,1]} \|f(x)-g(x)\|$ on the set $\operatorname{B}$ of uniformly bounded functions from the interval $[0,1]$ to $\mathbb{R}$, fix $g \in \operatorname{B}...
- 5,405
0
votes
1
answer
131
views
Is this a smooth approximation to the $\ell$-infinity distance actually a quasi-metric?
The $\|\cdot\|_{\infty}$-norm on $\mathbb{R}^n$ for $n\in \mathbb{Z}^+$ is not a smooth function. However, I came across this post which essentially says that a pointwise approximation to the maximum ...
- 169
0
votes
1
answer
223
views
Dense $G_{\delta}$ set with $\sigma$-porous complement is cofinite?
Let $X$ be a separable Banach space and $D\subseteq X$ be a
proper, connected, and dense $G_{\delta}$ subset of $X$,
$X-D$ is $\sigma$-porous.
Then is $X-D$ contained in a finite-dimensional ...
- 5,405
0
votes
0
answers
62
views
Terminology: maps which are bi-Lipschitz on compact subsets
Let $X$ and $Y$ be metric spaces and let $f:X\rightarrow Y$ be such that: for every compact subset $K$ of $X$ the restricted map $f|_K:K\rightarrow Y$ defined by $f|_K(x)=f(x)$ is bi-Lipschitz (with ...
- 5,405