I have a problem connected to Markowitz model. Through some calculations, my task comes down to finding extreme points. I have vector $a \in \mathbb{R}^4$, some $S$ and $V$, where $S$ - expected profit, $V$ - covariation matrix.
So, my problem is: how many extreme points can there be? I have example with $4$ extreme point and I have a guess, that there can't be more then $4$.
What I tried? I know that exist algorithm, which reduces the problem of extreme points to solving various linear equations. And therefore I can say that number extreme points less than $C_4^2 = 6$, but it's very rough estimate. I tried to show that for symmetry reasons there can't be $5$ points, but I failed.
So, now I am stuck on that problem and searching for any help.