I read a theorem stating that if $\Omega$ is a bounded open set in $\mathbb R^n$ and $\partial \Omega$ is $C^2$ then $\Omega$ satisfies exterior sphere condition. So can anyone tell me some counter-example where $\Omega$ is bounded and $\partial \Omega$ is $C^1$ but exterior sphere condition fails on some boundary point of $\Omega$.
Any type of help will be appreciated. Thanks in advance.
Consider a domain $\Omega \subset \mathbb R^2$ which coincides near the origin with the region $\{(x,y) : y < f(x) \}$ under the graph of $f(x) = |x|^{1+\alpha}$ for some $0<\alpha < 1.$ Since $f$ is $C^1$ but not $C^2,$ such a domain can be $C^1$ but not $C^2$, and is "too sharp" for any ball $B_r(x_0) \subset \mathbb R^2 \setminus \Omega$ to have $0 \in \partial B_r(x_0).$
To see this, first note that such a ball must be centred on the $y$-axis, since otherwise its boundary circle would have non-zero slope at the origin and thus cross into $\{y < 0\}\subset\{y < f(x)\}.$ Thus the boundary circle is the graph of the function $g(x) = r - \sqrt{r^2 - x^2}.$ In particular, note that regardless of $r>0$ we have $g(x) = O(x^2)$ as $x\to 0,$ so $g(x) / f(x) \to 0$ and thus for small $x$ we have $g(x) < f(x),$ implying the ball intersects $\Omega.$