In Adam's "Sobolev Spaces", he defines "segment condition" in 3.21 which describes the properties of a domain:
we say that a domain $\Omega$ satisfies the segment condition if every $x\in \text{bdry}\Omega$ has a neighborhood $U_x$ and a non-zero vector $y_x$ such that if $z\in \bar{\Omega}\cap U_x$, then $z+ty_x\in\Omega$ for $0<t<1$.
I'm confused about the definition above, in 3.20 he defines $\Omega_1:=\{(x,y)\in\mathbb R^2|0<|x|<1,0<y<1\}$ and says $\Omega_1$ doesn't satisfy the segment condition but $\Omega_2:=\{(x,y)\in\mathbb R^2|-1<x<1,0<y<1\}$ satisfies the segment conditon.
For $\Omega_2$, take $x=(0,1)$, then the only choice of $y_x$ in my mind is $(0,-\varepsilon)$. Take $U_x$ to be the semi-disc of radius $1/2$, then when $z$ is close to $(-1/2,1)$, certainly $z+ty_x\notin \Omega_2$.
the author concludes that " the domain cannot lie on both sides of any part of its boundary " and I don't know how to get it.