Fenchel–Moreau theorem
Okay kiddo, so there is a theorem called the Fenchel-Moreau Theorem (it's a big name, huh?). This theorem is all about something called convex functions.
Convex functions are like hills that slope up on both sides. Imagine you are on a hill and you start at the bottom. If you walk up either side of the hill, eventually you will reach the very top and the slope will start to go down on the other side. That's what a convex function looks like!
Now, the Fenchel-Moreau Theorem says that for any convex function, there is another function that you can pair it with called its "Legendre-Fenchel transform."
This Legendre-Fenchel transform is like a buddy that helps the convex function explain itself better. It tells you more about how the convex function is shaped and behaves.
So basically, the Fenchel-Moreau Theorem says that every convex function has a special buddy that helps you understand it better. Pretty cool, huh?
Convex functions are like hills that slope up on both sides. Imagine you are on a hill and you start at the bottom. If you walk up either side of the hill, eventually you will reach the very top and the slope will start to go down on the other side. That's what a convex function looks like!
Now, the Fenchel-Moreau Theorem says that for any convex function, there is another function that you can pair it with called its "Legendre-Fenchel transform."
This Legendre-Fenchel transform is like a buddy that helps the convex function explain itself better. It tells you more about how the convex function is shaped and behaves.
So basically, the Fenchel-Moreau Theorem says that every convex function has a special buddy that helps you understand it better. Pretty cool, huh?