# convexity and the interior sphere condition

Consider $\Omega$ a open, convex bounded subset of $R^n$. Let $x_0 \in \partial \Omega$. I believe that exists a open ball $B \subset \Omega$ such that $\partial B \cap \partial \Omega = \{ x_0 \}$. (this is the interior sphere condition in $x_0$) ? I have no idea how to proof ...

Someone can help me to prove or give a counter example for the question ?

• You have to ask some regularity on the boundary, in order that this result is true. For example, if $\partial\Omega\in C^1$, then this result is true. – Tomás Oct 29 '13 at 20:20
There is an easy counter example. $\Omega = int([0,1]^n)$ and $x_0 = (1,\ldots,1)$ then there is no ball which we want.
No. Let $\Omega$ be the standard open unit square. Let $x_0 = (1,1).$ There is no open disc contained in the open square for which $x_0$ is a boundary point.