Convex subset

Imagine you have a big bag of M&Ms. Now imagine you want to put some of those M&Ms on a plate in such a way that if you connect any two M&Ms with a straight line, the line will always be inside the plate. That means that the line will never leave the plate, like a maze that you can travel through without going outside the borders.

This plate with the M&Ms on it is actually a convex subset. "Convex" means that any two points inside it can be connected by a straight line that never goes outside the subset. In simpler words, it's like you draw a line between any two M&Ms on the plate and the line stays in the plate.

Another example would be a balloon. If you blow up a balloon, the surface of the balloon is convex. Any two points on the surface of the balloon can be connected by a straight line without leaving the surface.

In math terms, a convex subset is a subset of a space where, if you take any two points inside that subset, all the points lying on the straight line between them are also inside the subset. Therefore, like the M&Ms on the plate or the surface of the balloon, all of the points in the set are connected by a straight line to each other without ever exiting the subset.