I have the following Definition with I struggle with. Namely it is the definition of a open set of class $C^1$.
We say an open set $O$ in $\mathbb{R}^d$ is of class $C^1$ iff for sets
- $S_1:=\lbrace (x_1,\ldots,x_{d})\in \mathbb{R}^d : (x_1,\ldots,x_{d-1}) \text{ is an element of the open unit ball in }\mathbb{R}^{d-1}\text{ and }|x_d|<1\rbrace$,
- $S_2:=\lbrace x\in S_1 : x_d>0\rbrace$
- $S_3:=\lbrace (x_1,\ldots,x_{d-1},0)\in \mathbb{R}^d : (x_1,\ldots,x_{d-1}) \text{ is an element of the open unit ball in }\mathbb{R}^{d-1}\rbrace$
there exists for any $x\in \partial O$ a neighbordhood $U_x$ and a bijective map $B:U_x \rightarrow S_1$ such that
- $B \in C^1(\overline{O})$,
- $B^{-1} \in C^1{\overline{S_1}}$,
- $B(U_x\cap S_1)=S_2$ and
- $B(U_x\cap \partial O)=S_3$
Then $O$ is of class $C^1$. I have a hard time to imagine this. What are simple examples for such sets?
Also what I want to understand is, are such sets also uniformly Lipschitz open sets?
i.e. do they satisfy this Definition:
where
I also have a tough time understanding the intuition behind this definition. Examples are also welcome.
I feel like if we assume $\Omega$ in the second Definition to be bounded, then the first and second property in the definition of uniformly Lipschitz open can be dropped.
Then the definition looks very similar to the Definition in Evans:
Maybe one can help me to relate these three Definitions for open and bounded sets?