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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(\...
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 ...
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 ...
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 ...
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 ...
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, ...
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 ...
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 ...
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 ...
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}...
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+ \...