I've come across two definition and I cant understand the differences between them. Maybe someone can help me..Whats the difference between a lipschitz domain and a domain satisfying the cone condition? What are some counter examples?
Cone condition definition : $Ω ⊂ \mathbb{R}^n$ satisfies the cone condition if there exists a bounded cone with positive angle that can be moved within the domain to touch each boundary point such that the cone lies inside the domain.
Lipschitz domain definition:$Ω ⊂ \mathbb{R}^n$ has Lipschitz boundary if there is a finite open cover $\cup^n_{i=0}Ui\supset\bar{\Omega}$
such that for each i, $∂Ω ∩ Ui$ is graph of a Lipschitz continuous function.
The double brick domain in $\mathbb R^3$ is not Lipschitz but satisfies the cone condition.
It seems from the definition that if two domains satisfy the cone condition then their union will satisfy it as well, which is not true for Lipschitz domains.