punishment inflicted to minor children,
I suppose this means that the convention would prevent trying a minor as an adult under any circumstances.
This issue is often raised in the U.S. if the crime is violence against another child. Any UN convention involving sovereignty is going to be difficult (like the ICC).
One way to get us on board might be to show us this convention in action regarding sexual abuse of children by the third world nations. We are concerned that rules are not applied equally by the UN.