Is every compact hypersurface contained in a sphere which it touches twice?

Let $M\subset \mathbb{R}^{n+1}$ be a compact $n$-manifold. There exists, then, a smallest $n$-sphere containing $M$, and it must touch it in one point.

Must it touch it twice?

This seems quite intuitively right to me, but I've no idea how to prove it. It's easy to construct counterexamples where you can't have more than 2 (e.g. an ellipse which is not a circle).

Solution 1:

Yes. If not, let a smallest sphere containing $M$ touch $M$ at only $P$ and have center $O$. The sphere can be divided into two hemispheres one of which, the northern, say, has $P$ as north pole. The closed southern hemisphere, $S$, does not intersect $M$ so $\epsilon:=d(S,M)>0$. Translate the sphere $\epsilon/2$ in the direction $OP$. The translate of $S$ is still distance $\epsilon/2$ or more away from $M$, so it does not intersect $M$. The translate of the closed northern hemisphere is contained in the region formerly outside the sphere, so it does not intersect $M$ either. Therefore the translated sphere is not smallest, so neither was the original sphere.