In this paper They have defined the assumption (H) for an open $\Omega$ in $\mathbb R^d $ as:
(H) $\Omega$ is a bounded, open, star-shaped domain with respect to $x_0\in \operatorname{int}(\Omega)$, $B(x_0,\epsilon)\subset \Omega$, for some $\epsilon>0$, there exists a finite partition of $\Omega$, $\Omega =\Omega_1\cup..\cup\Omega_p$, such that $\Omega_i$ is a cone with vertex at $x_0$, $\partial \Omega\cap\partial\Omega_i\in\mathcal C^1$, $B(x_0, \epsilon)\cap \Omega_i$ is convex for all $i = 1, ..., p$. Further, we assume that there exist $a > 0$ such that $v(x).x > a$ for all $x\in\partial\Omega$, where $v(x)$ denotes the outward unit normal to $\partial\Omega$ at $x$.
My question is: if $U$ is a Lipschitz bounded, open, star-shaped domain, Is what $U$ satisfies (H).
Note that: We say that $\Omega$ is star-shaped with respect to an interior point $x_0\in \Omega$ , if any ray with origin at $x_0$ has a unique common point with the boundary $\partial \Omega$.