Let $$\Sigma = \left\{ (x_1,\dots, x_n) \in \mathbb{R}^n : \sum_{i=1}^n x_i^2 = 1 \right\}$$ and let $S$ be a finite subset of $\Sigma$ and let $C$ be the convex hull of $S$. Show that any point of $S$ is an extreme point of $C$.
So it seems we need to show that :
$ \forall x \in S \subset \Sigma , \: \: \forall [y,z] \subset C $,
$ x \in [y, z] \Rightarrow x = y \: $ or $ \: x= z $
We know that $ ty + (1-t)z, t \in [0,1] $ can be written as $ \sum \lambda_i s_i, \sum \lambda_i = 1, \lambda_i \geq 0 $, $ s_i \in S$
But why if $ x = ty + (1-t)z $ then necessarily t $ \in \{0,1 \} $ ?
Also C looks like a polytope to me.
Any help on this would be great.
If for some $x\in S$ there are $y,z\in C$, $y\neq z$ and $t\in(0, 1)$ such that $x = ty + (1 - t)z$, then we also have $$\langle x, x\rangle = 1 = t \langle y, x\rangle + (1 - t) \langle z, x\rangle.$$
But either $y$ or $z$ is different from $x$, hence either $\langle y, x\rangle$ or $\langle z, x\rangle$ is strictly smaller than $1$. This leads to a contradiction.