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.




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> **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 $$F(x) = \min \left\{ f(x,y) ; y \in \mathbb{R}^n \right\}$$ is well defined, i.e., if the minimum always exists then $F$ is always strictly convex.
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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.



Convexity of a minimum function
This is a standard result in convex analysis. See for example, $\S$3.2.5 of [*Convex Optimization*](http://www.stanford.edu/~boyd/cvxbook/) by Boyd and Vandenberghe (just slightly modify their proof to conclude strictness).


