Decompostition of a Lipschitz domain We say that $\Omega$ is a strongly star shaped domain (with respect to $0$ for example) in $\mathbb R ^n$ if: 
$$\Omega  = \{x\in \mathbb R ^n : \left \| x \right \| < g(\frac{x}{\left \| x\right \|})\} $$ and 
$$\partial \Omega  = \{x\in \mathbb R ^n : \left \| x \right \| = g(\frac{x}{\left \| x \right \|})\} $$
with $g$ is a continuous, positive function on the unit sphere.
In this paper,
Bramble uses the fact that : Any Lipschitz domain can be written as the union of strongly star shaped Lipschitz domains: $\Omega=\cup_{i=1}^{M}\Omega_i$
Can you help me to find why do we have this result? Do you have any references in which I can find this proposition?

PS: Sorry to ask this question again, but this did not get answered (as I had desired) on M.SE.
 A: You may like to check such results, in particular, Proposition 2.5.4 of the below monograph. Hope you have access to it.

Proposition 2.5.4. Let $\Omega \in \mathcal A_0$ have Lipschitz boundary. Then there exists a finite open covering  $\{\Omega_j\}_{j\in\{1,\dots,m\}}$ of $\overline\Omega$ such that, for every $j\in\{1,\dots,m\}$, $\Omega_j \cap \Omega$ is strongly star shaped with Lipschitz boundary. 

Carbone L. and De Arcangelis R., Unbounded functionals in the calculus of variations: Representation, relaxation, and homogenization,
Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics,
vol. 125, Chapman & Hall/CRC, Boca Raton, FL, 2002
A: A Lipschitz domain $\Omega$ is an open set and any open set is a union of balls, which are strongly star-shaped. So I assume you meant $\overline\Omega$. By definition, any point of $\partial\Omega$ has a nbd in $\overline\Omega$ which is isometric to a sub-graph of a positive $k$-Lipschitz function $f:B(0,r)\to(0,+\infty)$,  $$\{(x,t)\, :\, |x|<r, \;0<t<f(x)\}.$$
We can take  $r<{f(0)\over 2k+1}$, which makes the latter set strongly star-shaped w.r.to the point $(0,r)$ as it is easy to check. 
