How can I solve this max min problem using CVX?

Have you proven this is a convex optimization problem?

min(convex) is not allowed by CVX, because its convexity or concavity can’t be determined as such.

Also note that you need to assign an expression const, before using it, as you did in the maximize statement. because you did not, it has the default value of zero at the time of its appearance in the maximize statement; and therefore is the maximizing 0, which is the same as not having an objective at all.

Also, you have changed the norm squared constraint to <= from `, which is needed to make that constraint convex, but is not necessarily equivalent.

I will mark this as a non-convex problem, unless and until you prove otherwise.