Let $\Omega\subset\mathbb{R}^N$ be an open and connected Lipschitz domain. Consider an $N$-dimensional open ball $B(x,r)$ that intersects $\Omega$. Is it true that $\Omega\cap B(x,r)$ is a Lipschitz domain too?

It is obvious if $B(x,r)\subset\Omega$, but what happens in the remaining case?

**More general question: The intersection of two bounded Lipschitz domains is also Lipschitz?** 

I don't know any reference that specify those facts and I decided to ask here.


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  [1]: https://i.sstatic.net/GPf5V6sQ.png