I am reading the following paper: https://docs.wixstatic.com/ugd/1de1d9_cd82cb002eaa4eefa9af574eb5efdff2.pdf
I am stuck on the proof of lemma 2.3 on page 6.
I don't see why does the property (i) of Lipschitz domain is satisfied? (it's listed on the bottom of page 5), what is $\delta:=\delta(d,R)$ explicitly?
I don't understand the last paragraph:
Now, for any $x\in \partial \Omega \cap U_j$, the cone defined by the convex closure of $x\cup B(0,1)$ is contained in the closure of $\Omega$. Any head angle $\alpha$ of this cone satisfies $\sin(\alpha/2)\ge 1/R$ and thus the boundary is $Lip 1$ function with $M:=M(d,R)$.
Can you elaborate on the two sentences in the quote above? and with the specifics, cause I don't see why the convex closure of $x\cup B(0,1)$ cannot exceed the closure of $\Omega$; I also don't see how did they arrive at the inequality in the last sentence of $\sin(\alpha/2)$, perhaps I am missing something from elementary geometry. why is the boundary $Lip1$?
Thanks.
