Overlapping hemispheres [closed]

Suppose that $d$ points are selected on the surface of a sphere.

Each point "induces" a hemisphere of the sphere. That is, it defines a hemisphere "centered" at that point (i.e. the "top" / "pole" of the hemisphere is at that point).

Prove that all $d$ points are located within one same hemisphere $\iff$ the $d$ induced hemispheres all overlap somewhere.

Thank you.

Let $a_1\dots,a_d$ be your $d$ points and $R$ the radius of the sphere. Thus $\|a_i\|=R$.

Then by definition, a point $x$ is on the hemisphere defined by $a_i$ if and only if $$\langle x, a_i\rangle \geq 0 \text{ and }\|x\|=R\tag{1}$$

The nice thing with this property is that it is symmetric in $x$ and $a_i$: Saying that $x$ is on the hemisphere whose pole is $a_i$ is equivalent to saying that $a_i$ is on the hemisphere whose pole is $x$.

Now, applying that property $(1)$, to say that all $d$ points are located on the same hemisphere (e.g. whose pole is some point $u$, with $\|u\|=R$) is equivalent to saying that $\langle a_i, u\rangle\geq 0$ for all $i$. And by $(1)$ that's equivalent to saying that $u$ itself is on the hemisphere defined whose pole is $a_i$, and that, for all $i$. And this is equivalent to saying that point $u$ is in the intersection of all those hemispheres.

Note that we proved more than what you were asking. We proved that the overlap is exactly the set of poles of hemispheres that contains all $d$ points.