I am trying to push the mindset that entropy is just a tool to do convexity arguments clearly.
A good milestone would be if one could prove Jensen's inequality (for finite random variables etc.) via entropy.
Of course properties in entropy use Jensen, I don't care about that.
You're allowed to use however powerful tools you want (i.e maybe a proof would need to approximate a convex function with simpler function we can connect to entropy).
Thanks