Invariant convex cone

An invariant convex cone is like a special kind of bucket that always keeps its shape and can only hold certain types of things inside.

Imagine you have a bucket that is shaped like a cone, but instead of being made of plastic, it's made of a special invisible material. This bucket only allows you to put things inside that are nice and spread out evenly, without any sharp edges or corners. Also, no matter how you rotate or turn the bucket, it always stays the same shape.

Now, imagine you have a bunch of colored balls. Some of these balls are much heavier than others, and some are closer together than others. But because of the special cone-shaped bucket, you can only put the balls inside that meet certain requirements - they have to be spread out evenly, without any sharp edges or corners, and they have to be nice and balanced.

This special kind of bucket is called an invariant convex cone, and it's used in math to describe a particular type of set. The idea is that this set has certain properties that always stay the same, no matter what you do to it (invariant). It's also shaped a certain way, like a cone, and it only allows certain types of things inside (convex).

So, if you hear someone talking about an invariant convex cone, just think of it like a special bucket that only allows certain things inside, and doesn't change shape no matter how you turn it.