# Lipschitz smooth boundary definition

Is the wikipedia definition of Lipschitz Euclidean domain correct?

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?

• What stops you is that the scalings and translations might not satisfy what they would need to. – user5810 Oct 29 '10 at 5:58
• For each point p there is a corresponding radius r>0. Would I be allowed to scale and translate to x within the domain of h_p shrinking the radius about x appropriately? – alext87 Oct 29 '10 at 6:11
• I can't come up with anything other than a circle that you could do that for. (try it with a corner and the midpoint of an edge from a square) – user5810 Oct 29 '10 at 6:24
• I'm completely lost by the definition so I seem to get that it works. Why is it working for a circle? – alext87 Oct 29 '10 at 7:04
• I am retagging this classical analysis. Euclidean domain does not mean what you think it means: en.wikipedia.org/wiki/Euclidean_domain – Willie Wong Oct 29 '10 at 9:06

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.