Uniform cone property - A domain $\Omega$ is said to have the uniform cone property if there exists a locally finite open cover $\{U_j\}$ of $\partial \Omega$, and a corresponding sequence $\{C_j\}$ of finite cones such that :
(1) For some finite $M$, every $U_j$ has diameter less than $M$.
(2) For some $\delta >0$, $\Omega_{\delta}\subset\cup_{j=1}^{\inf} U_j$. Where $\Omega_{\delta}=\{x \in \Omega:dist(x,\partial\Omega)<\delta\}$.
(3) For every j, $\cup_{x \in \Omega \cap U_j}(x+C_j)=Q_j\subset \Omega$. Where $x+C_j$ is the translation of $C_j$ by $x$.
(4) For some finite $R$, every collection of $R+1$ of the sets $Q_j$ has empty intersection.
Strong Lipschitz property - A domain is said to have this property if there exists positive numbers $\delta$ and $M$, a locally finite open cover $\{U_j\}$ of $\partial \Omega$, and for each $U_j$ a real-valued function $f_j$ of $n-1$ real variables, such that the following conditions hold:
(1) For some finite $R$, every collection of $R+1$ of the sets $U_j$ has empty intersection.
(2) For every finite pair of points $x$, $y\in \Omega_{\delta}$ such that $|x-y|<\delta$ there exists $j$ such that
$x,y \in \{x\in U_j:dist(x,\partial U_j)>\delta\}$.
(3) each function $f_i$ are Lipschitz.
(4) For some coordinate system $(x_1,x_2,...x_n)$ in $U_j$ the set $\Omega\cap U_j$ is represented by $x_n < f_j(x_1,...x_{n-1})$.
These are the definitions given in the book on Sobolev Spaces by R.A Adams.
I am not able to understand how to interpret these properties geometrically and moreover how the Strong local Lipschitz property implies the uniform cone property. An example of a strong locally Lipschitz domain in $\mathbb{R^2}$ will be really helpful.