Convex metric space

A convex metric space is a special kind of space where everything is like it's made of squishy jello. Imagine you have a bunch of different points, kind of like little bugs, and you can move them around, but they have to stay squished together like jello.

In a convex metric space, you can measure how far apart these jello-bug-points are from each other. But there's a special rule: you can't measure distance by cutting through the jello-bugs. Instead, you have to find a path that stays inside the jello, like a worm crawling through jello.

Now, imagine you have two jello-bugs that are far apart. You can squish the jello-bugs closer together by moving them along the worm-like path, but they always have to stay in the jello. This means that you can always find a worm-like path that connects any two points in a convex metric space.

Overall, a convex metric space is a space where everything is made of squishy jello, and no matter how far apart two jello-bugs are, you can always find a worm-like path that connects them while staying inside the jello.