I am trying to prove the following thing that in 2 or 3 dimensions seems intuitive:
In $\mathbb{R}^n$, let $\varphi$ be a hyperplane passing by the point $Q$ and let $\mathbb{S}$ be a hypersphere, such that $\varphi$ is tangent to $S$ at $Q$. Let $A$ is a point which is not in the hyperplane and it is outside the hypersphere (so not inside or on the border). Then if the segment $\bar {AQ}$ does not intersect the hypersphere (in any other point but Q), then $A$ is not in the same half space as the hypersphere $\mathbb{S}$. How can I do that? I have tried it for hours but I don't have anything worth writing here.
If the line passing trough $A$ and $Q$ only intersects the closed ball at $Q$ then the line is tangent to the sphere and is therefore inside the hyperplane $\varphi$.
It follows that if $A$ is not on the hyperplane then the intersection of the line $AQ$ with the (closed) ball is not a single point $Q$. In particular since this intersection is an intersection of closed convex sets it is itself a closed convex set. Therefore the intersection is a line segment containing $Q$ and a subset of the line $AQ$
In particular if the segment $\overline{AQ}$ does not intersect the hypersphere we know the intersection of $AQ$ with the ball is a line segment $\overline {QA'}$ where $A'$ is on the line $AQ$ and in the closed ball. Clearly $A$ and $A'$ are not in the same half-space. Since $A'$ is in the same halfspace as the sphere it follows that $A$ must not lie in the same halfspace as the sphere.