Line Segment Conditions

Definition¹

Let $\Omega \subset \mathbb{R}^{n}$ be an open set. For all $x \in \mathrm{bdry}\Omega$, if there exists a neighborhood $x$ of $U_{x}$ and a nonzero vector $y_{x}$ such that the following condition is satisfied, then $\Omega$ satisfies the segment condition.

$$z\in \overline{\Omega}\cap U_{x} \quad \implies \quad z+ty_{x} \in \Omega, 0 \lt t \lt 1$$