# Convexity of a minimum function

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

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.

• I gave a reference because that book is freedly downloadable from the link provided. It's strange to explicitly forbid a reference though. Commented Sep 7, 2013 at 15:07

This is a standard result in convex analysis. See for example, $\S$3.2.5 of Convex Optimization by Boyd and Vandenberghe (just slightly modify their proof to conclude strictness).
An idea of the proof. For the convexity of $$F$$: a function is convex iff its epigraph is convex; the epigraph of $$F$$ is the projection of the epigraph of $$f$$; the projection of a convex set is convex. Note that this part also work with $$\inf$$ more generally that $$\min$$ in the definition.
For the strict convexity: assume that $$F$$ is convex but not strictly convex. Then, up to adding a linear form to $$F$$, the $$F$$ has more than a minimum point. Any minimum point of $$F$$ is the projection of a minimum point of $$f$$, and since $$F$$ has more than a minimum point, so does $$f$$, and $$f$$ is not strictly convex.
Finally, note that a strictly convex and bounded below function $$f$$ does not produce in general a strictly convex $$F(x):=\inf_y f(x,y)$$, like the example of $$f(x,y):=\exp(x^2/2+y)$$ shows, for $$f$$ is strictly convex and $$F$$ is identically $$0$$.