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Let $\|\cdot\|$ be a norm on $\mathbb{R}^n$. Let $C:=\{x\in\mathbb{R}^n\,:\,\|x\| \leq 1\}$, that is to say let $C$ be a convex compact symmetric set of non empty interior. Let $H$ be a linear subspace of $\mathbb{R}^n$ of dimension $n-1$. Is it true that there exists $z\in \mathbb{R}^n$ such as $\|z\|=1$ and \begin{align*} \Big[(H \cap \partial C) + \mathbb{R}z\Big] \cap \mathring{C} = \emptyset \quad\text{ ? } \end{align*} If not, please, give me a counter-example ! It appears to me that $z\in H^\perp$ should be a proper candidate, but though I might see why this is working for the $\| \cdot \|_p$ norms for example in low dimensions, I did not manage to prove it generally.

Thanks in advance.

Xou.

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    Did you mean $\leq$ in the definition of $C$?2012-09-26
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    What's for you $\,\Bbb Rz\,$??2012-09-26
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    @DonAntonio: Presumably $\{\lambda z \}_{\lambda \in \mathbb{R}}$?2012-09-26
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    what is $\mathring{C}$?2012-09-26
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    Most likely the interior.2012-09-26
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    Presumably you mean $z \neq 0$, since the result is trivially true in that case?2012-09-26
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    Yes sorry, I meant $\leq$ and $z\neq 0$ (I corrected both). As for the two other questions, the given answers are correct !2012-09-26

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