Q: the boundary normal vectors Suppose $X, Y$ are convex compact sets in $\mathbb R^n$ and $z \in  \mathbb R^n$. 
Would it be the case that
$$\max_{x\in X}  [ \langle x, z \rangle-\max_{y\in Y} \langle x  ,y \rangle ]=\max_{x\in X} \min_{y\in Y} \langle x, z-y \rangle =\min_{y\in Y} \max_{x\in X}  \langle x, z-y \rangle $$
In this way, I may calculate the  $x^*$ (a subclass of normal vectors in $X$) that solves
$$\max_{x\in X}  [ \langle x, z \rangle-\max_{y\in Y} \langle x  ,y \rangle ]$$
Thanks.

A: Your claim is true. The first equation follows from the $-\max(\cdots)=\min(-\cdots)$. The second follows from a generalization of the minimax theorem. For example you can use Sion's minimax theorem. All the conditions on the function $f$ are satisfied trivially in your case because your function is linear.
