Why a $C^2$ domain satisfies the interior ball condition? I accept a reference too. Thank you.
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Could you please add some context? – egreg Apr 06 '14 at 23:34
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https://math.stackexchange.com/questions/284562/interior-sphere-condition?rq=1 – Guy Fsone Jul 18 '22 at 10:46
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https://math.stackexchange.com/questions/43169/smooth-boundary-condition-implies-exterior-sphere-condition – Guy Fsone Jul 18 '22 at 10:49
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Does this answer your question? Smooth boundary condition implies exterior sphere condition – Guy Fsone Jul 18 '22 at 10:50
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Have you found a reference ? I would like that. – Lucas Linhares Aug 25 '22 at 23:53
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Take the function $\psi$ from your previous question, A question about $C^2$ domain., and use its Taylor expansion of second order to obtain an upper bound $\psi(x)\le C|x|^2$ for small $x$. This will ensure that for small $r$, the sphere of radius $r$ centered at $re_n$, stays above the graph of this function.
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Can you explain why does $\Psi(x)\leq C|x|^2$ for small $x$ imply that for small $r$, the sphere of radius $r$ centered at $re_n$, stays above the graph of this function? Thanks! – Xianjin Yang Apr 02 '17 at 16:11