Is the wikipedia definition of Lipschitz Euclidean domain correct? 


See: <http://en.wikipedia.org/wiki/Lipschitz_domain>


i was wondering what stops me just showing the condition holds for one point and then just scale and translate that function $h\_p$ for any point on the boundary... This doesn't seem right? What am I missing?


Thanks in advance.



Lipschitz smooth boundary definition
A domain of $\mathbb{R}^n$ with Lipschitz boundary is an open subset $\Omega\subset \mathbb{R}^n$, which is locally the sub-graph of a Lipschitz function w.r.to some choice of orthogonal coordinates. In other words, for any $p\in\partial \Omega$, up to an orthogonal change of coordinates, there is an open set 
$V\subset\mathbb{R}^{n-1}$, and a Lipschitz function $\phi:V\to(a,b)$ such that $U:=V\times (a,b)$ is a nbd of $p$ and 
$$\Omega\cap U=\{(x,t)\in U\ | \ t< \phi(x) \}.$$ 
This is equivalent to the definition given in the link, which is closer to the general definition of manifold with boundary (here the transition mappings are the bi-Lipschitz homeomorphisms). 


In particular, for any point $p\in \partial\Omega$ there is a small cone 
$C$ with vertex in the origin, and a nbd $U$ of $p$ such that for any $q\in U\cap\partial\Omega$ the cone $q+C$ is disjoint from $\Omega$, and the cone $q-C$ is included in $\bar \Omega.$ This is a third equivalent definition.


