Lipschitz smooth boundary definition 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.
 A: 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.
A: The definition given in the link is sometimes called "weakly Lipschitz".
The two definitions given by Pietro Majer are indeed equivalent, they are sometimes called "strongly Lipschitz" and "strong cone condition", respectively.
If the boundary of the domain is compact, then any cover of the boundary has a finite subcover. So in all the definitions, it doesn't matter whether we state the property for a neighborhood of every point, or for a finite cover.
One can easily see that strongly Lipschitz implies weakly Lipschitz.
However, the converse is not true. This has to do with the fact that the implicit function theorem does not hold for Lipschitz continuous functions.
E.g., there exist polyhedra that are are weakly Lipschitz but not strongly Lipschitz; they fail to be a graph of a Lipschitz continious function.
For more see P. Grisvard, Elliptic Problems on Nonsmooth Domains, Pitman, Boston, London, Melbourne, 1985.
Unfortunately, in the literature "Lipschitz" is occasionally used meaning "strongly Lipschitz", but sometimes meaning "weakly Lipschitz".
