[oersted]

My first thought is that it's rather obvious, but I'm probably wrong, can you help me understand?

The analogy I have in mind is: if you throw n dice, for large n, the likelihood of one specific chosen dice being high value and the rest being low value is obviously rather small.

I guess that the consequence is still interesting, that most random points in a high-dimensional n-sphere will be close to the equator. But they will be close to all arbitrary chosen equators, so it's not that meaningful.

If the equator is defined as containing n-1 dimensions, then as n goes higher you'd expect it to "take up" more of the space of the sphere, hence most random points will be close to it. It is a surprising property of high-dimensional space, but I think it's mainly because we don't usually think about the general definition of an equator and how it scales to higher dimensions, once you understand that it's not very surprising.

[isoprophlex]

> The analogy I have in mind is: if you throw n dice, for large n, the likelihood of one specific chosen dice being high value and the rest being low value is obviously rather small.

You're exactly right, this whole thing is indeed a bit of an obvious nothingburger.