I would like to prove there exists $C^{1,\alpha}$ with $0<\alpha<1$ not satisfying the interior Sphere condition. I consider $\Omega=\{(,)\in \Bbb R^2:>||^{1+\alpha}\},$ with $0<\alpha<1$. This domain is $C^{1\alpha}$ smooth. From the graph of $y=|x|^{1+\alpha}$ I can see that $(0,0)\in \partial \Omega$ does not satisfies the interior sphere condition.
How can I prove this formally?
Recall Definition: A point $a\in \partial \Omega$ satisfies the interior sphere condition for $\Omega$ if there exists a ball $B,$ $B\subset \Omega$ such that $\partial \Omega\cap \partial B=\{a\}$.