Fenchel duality

Okay, let's imagine you and your friend want to trade some toys. You have a teddy bear and your friend has a Barbie doll. Both of you want to make a fair trade, so you need to figure out how much each toy is worth.

Fenchel duality works kind of like this toy trade. It helps you figure out how much two different things are worth by comparing them to each other in a fair way.

In math, we call the two things we're comparing "functions." A function is like a recipe that tells you how to cook something. If we have two different recipes, we want to know how much they're worth compared to each other.

To figure this out, we use something called the Fenchel transform. This is like a magic wand that turns one recipe into another recipe that's easier to compare. The new recipe is called the "dual" of the original recipe.

Just like with the toys, we want to make sure the trade is fair. So we use the Fenchel duality to make sure both recipes are worth the same amount. If one recipe is more valuable, we know we need to add more to the other recipe to make things even.

Overall, Fenchel duality helps us make fair trades between different functions. It's a very useful tool for mathematicians who work with lots of different recipes (functions) and want to compare them fairly.