Apologies if I am missing something obvious here. I am trying to understand counterexamples to the segment property. From reading Adams' book titled Sobolev Spaces, the definition is given as follows:
Definition: We say that a domain $\Omega$ satisfies the segment condition if for every $x \in \partial\Omega$ has a neighborhood $U_x$ and a nonzero vector $y_x$ such that if $z \in \overline{\Omega} \cap U_x$, then $z+ty_x\in \Omega$ for $0<t<1$.
After the definition, it states that "the domain [satisfying this condition] cannot lie on both sides of any part of its boundary." A similar geometric intuition is given in segment condition in sobolev space. But I don't understand why this intuition works at all. If I take a domain that is kidney shaped, or I take an annulus, this seems to satisfy the condition that Adams and the Stack question are proposing as types of domains that wouldn't have the segment property. However, it seems pretty self-evident that both of those domains have the segment property. The reason the counterexample domain in the stack question works is because the boundary is enclosing a point of $0$ Lebesgue measure. Any help understanding the general intuition of categorizing domains that don't have the segment property in the way Adams and the Stack question seem to intuit would be greatly appreciated.
