Hyperplane intersecting a compact set at exactly one point? Let $E$ be a compact subset of $\mathbb{R}^n$. Does there always exist a hyperplane that has exactly one point of intersection with $E$? 
If not, what is a counterexample? And are there additional properties of $E$ that would make the statement true?

Let $E\subset \Bbb R^n$ be a non empty compact set. The function $f:E\to [0,\infty)$ defined as $f(x)=d(x,0)$ meets its maximum at some point $M$. The hyperplane orthogonal to the vector $\overrightarrow{OM}$ that contains $M$ has its points strictly farthest from $O$ than $M$, except for $M$ itself. 
