Minimax criterion in decision theory

Question

I have the following risk functions and I want to apply minimax criterion to both of them.

1. $inf_c sup_x$ ($c^2+(c-1)^2x^2$)
2. $inf_c sup_x$ ($xc^2+(c-1)^2x^2$)

For the first case I think that the minimax rule is for $c=1$, and for the second case I think that the minimax rule does not exist.

However I am not sure about my calculations, and so I need some help. Also, I need some guidelines how to work on such problems.

Answer 1

If by minimax rule you mean an optimal solution for the outer inf, then yes, for 1, when c=1, the objective function $f(c):=\sup_x c^2 + (c-1)^2 x^2$ takes the value $f(1)=1$, and $f(c)=+\infty$ for $c\ne 1$, so $\inf_c f(c)=1$ at $c=1$.

For 2, $g(c):=\sup_x c^2 x + (c-1)^2 x^2$ always has $g(c)=+\infty$ for all $c\in\mathbb{R}$. So $\inf_c g(c)=+\infty$. $g$ in this case is not a proper convex function.