I've been thinking of ways of making some sort of rigorous analogue (in the sense of obeying a small set of properties) to higher dimensional manifolds, like spheres and cubes.

For example, an unit sphere inscribed inside a high dimensional unit cube looks a lot like an astroid[1] inscribed in a square in 2D with regards to metric properties (using an L^2 metric):

- The astroid touches the cube, while it is very far to the cube in the direction of the corner

- We can see that the corners, being far from the unit-L^2 distance manifold, occupy most of the volume

- As the number of dimensions increase, the astroid becomes increasingly compressed near the origin

[1] https://upload.wikimedia.org/wikipedia/commons/thumb/0/03/As...