I was reading a proof of $9g-9$ theorem which states that $9g-9$ length parameters are sufficient the parametrize the Teichmuller space of a closed surface of genus $g$. The proof uses the following fact.

Theorem: Let $f:\mathbb{R}^m\times \mathbb{R}^n\rightarrow \mathbb{R}$ be a strictly conves function. If the function $F:\mathbb{R^m}\rightarrow \mathbb{R}$ is defined by $$F(x)=min\{f(x,y);y\in\mathbb{R}^n\}$$ is well defined, i.e. if the minimum always exists then $F$ is always strictly convex.

Can someone please give me any proof or at least idea of the proof of this fact.

P.S: I am reading the book "A primer on mapping class group." And I don't want a reference.