Convex conjugate

Imagine you have a jelly sandwich. Now imagine you take the crust off and flatten the sandwich so it becomes a straight line. The jelly on one side represents a function, and the jelly on the other side represents its convex conjugate.

Now, let's pretend you have a friend who is really good at math. They come over and take a look at your sandwich line. They start to draw some lines on it and at some points, they draw a line so that it just grazes the jelly. This line is called a tangent line.

The tangent line is really important because it tells you the steepness of the jelly. The steeper the jelly, the higher the tangent line will be. The lower the jelly, the lower the tangent line will be.

Now, here's where things get a little tricky. In math, we like to find the opposite of things. So, instead of finding the tangent line, we want to find the line that is opposite to it. This line is called the normal line.

In terms of the jelly sandwich, the normal line represents the convex conjugate. It's like the mirror image of the tangent line. The normal line is really helpful because it tells you something else about the jelly. It tells you how "curvy" the jelly is.

So, to recap: a convex conjugate is like the opposite of the steepness of a function, represented by the normal line drawn on the jelly sandwich. It tells you how "curvy" the function is.