I want to prove that the convex hull of a region defined by the following inequality is convex:
$$(y-z)^2 -4x w \geq 0$$,
where $x + y + z + w = 1$ and $x, y, z, w \geq 0$. I tried using the Hessian matrix and checking if it's positive semi-definite throughout the domain, but I was unsuccessful. Does anyone have any hints?