I am reading a proof of a lemma in Evan's book on PDEs (Pg.126), where he makes the following assertion:
$$\max_{z}\left\{\text{Lip}(g)|z|-L(z)\right\}=\max_{w\in B(0,\text{Lip}(g))}\max_{z}\{w\cdot z-L(z)\}.$$
Here $\text{Lip}(g)$ is the Lipshitz constant for the function $g,$ $L$ is a convex function which is actually the Lagrangian and $z\in \mathbb{R}^n.$ I am not sure why this true. Perhaps someone can explain?
There's nothing deep here. Let me denote $C:=\mathrm{Lip}(g)$. Then, obviously, $$ C\lvert z\rvert = \max \left\{ w\cdot z\ :\ w\in\mathbb R^n,\ |w|\le C\right\}.$$ Therefore $$ \max \{C|z|-L(z)\ :\ z\}=\max \left\{ \max_{|w|\le C}w\cdot z - L(z)\ :\ z\right\},$$ and the most internal "max" is independent of $z$, so you can bring $L(z)$ inside of it, yielding $$ \max_z \left\{\max_{|w|\le C} \{ w\cdot z - L(z)\}\right\},$$ which is what the book says.