Understanding inf$_zF(z)$

Question

I am trying to understand the proof of "The set if minima of F is convex where F is a convex function)

The proof starts with considering $ F(x) = F(y) = \inf_z F(z) $ and then shows the contradiction of $ F(tx + (1-t)y) \le tF(x) + (1-t)F(y) = \inf_zF(z) $ for all $ t \in (0,1) $

I not getting the idea of $\inf_zF(z)$ component of this proof. Can you share insight please.

Answer 1

I think it is easier to show that the level sets are convex.

Let $L_\alpha = \{z | F(z) \le \alpha \}$. If $z_1,z_2 \in L_\alpha$, then $f(\lambda z_1+(1-\lambda) z_2) \le \lambda_1 F(z_1)+ (1-\lambda) F(z_2) \le \alpha$ and so $L_\alpha$ is convex.

Now let $\alpha = \inf_z F(z)$. Then $L_\alpha$ is convex. (Note that $L_\alpha$ may be empty, for example, let $F(z) = e^z$, then there is no minimum, but $\inf_z F(z) = 0$.)