Can't proove that a function is convex

Question

I'm trying to proove that a couple of functions are convex but one of them is giving me a hard time. Here it comes :

I can't see how this function can be written as the $\max$ or $\sup$ of some family of functions so that I can use the hint. Any ideas ?

Thanks !

Answer 1

Note that $y^T X^{-1} y = 2 \sup_x ( y^T x - {1 \over 2} x^T X x ) $.

Note that for a fixed $x$, the function $(y,X) \mapsto y^T x - {1 \over 2} x^T X x$ is linear, hence convex.

Hence the function $(y,X) \mapsto y^T X^{-1} y$ is convex.