As one increases more dimensions, it becomes increasingly likely that at least one of them will be an "extreme" value (i.e. point is at the edge of the hypercube ).

In the random() * random() * random() ... , once you draw at least one number close to 0 (chances will increase as you add more randoms), then your output collapses to a very small number. And in this visualization, 0 is shown as a point at the edge of the circle.

It depends upon what metric you use to measure it.

Of course with Euclidean it will as that seems to match our expectations for low dimensional space.

I'm not a mathematician so there is not much more that I could say on the matter at this juncture.