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Consider the subset $\Omega\subset\mathbf{R}^n$, $\Omega=\{x=(x_1,...,x_n)\in\mathbf{R}^n;x_n>\varphi(x_1,...,x_n)\},$ where $\varphi$ is a Lipschitz continuous function, that is, $\Omega$ is a unbounded set, bounded for a Lipschitz graph.

Why this set satisfies the interior sphere condtion?

Interior sphere condition means that for each $z\in\partial\Omega$, there is a ball $B_r(\xi)$ satisfying $\partial B_r(\xi)\cap\overline\Omega=\{z\}$.

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    As far as I can tell, this is straight analysis, so I've removed the differential-geometry and pde tags.2012-11-22

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This set does certainly not satisfy the interior sphere (or interior ball) condition in general, as the example $\varphi (x) = |x_1| + \ldots + |x_{n-1}|$ shows. This functions is $1$-Lipschitz, but the condition is violated at $0$.

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    Thank you very much Lukas Geyer.2012-11-22