0
$\begingroup$

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 ?

thanks in advance

$\endgroup$
  • 1
    $\begingroup$ 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. $\endgroup$ – Tomás Oct 29 '13 at 20:20
2
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.