Let $\Omega $ be an open, connected, bounded subset of the Upper Half space $ \Bbb R^n_+$ with a $C^{2,\epsilon}$ boundary. Then if its boundary $\partial \Omega$ (taken with respect to $\Bbb R^n$, i.e. not just with respect to the upper half space!) contain the origin, does there exist $\delta>0$ such that $B(0,\delta) \times \{0\} \subset \partial \Omega$ (note here that $B(0,\delta)$ is a ball of $\Bbb R^{n-1}$)
When I try to think geometrically, if it contains such a ball, as $\Omega$ is bounded, the boundary has to lift from the $\Bbb R^{n-1}$ somewhere and there we are losing the smoothness! So I think the answer to the question should be no. Though I can't write my idea rigorously.
Can someone please help.