It’s a statement about your small neighborhood when you at the surface of the n-d sphere. And in low n it’s like half inside the sphere. In high n it’s more like a needle touching your position.
I’m not sure how that ^follows from the fact that most of volume of a high-dimensional sphere is at the outer surface.
The needle effect occurs because there’s more volume just above the surface than just below it.
EDIT: If you imagine expanding the large sphere slightly to encompass the entire small sphere, most of the volume of the new large sphere will be outside of the original large sphere.
If you imagine expanding the large sphere slightly to encompass the entire small sphere, most of the volume of the new large sphere will be outside of the original large sphere.
Well, if you are on the face on n-d cube, half of your neighborhood would be inside, no matter what n. But “most volume in the peel” is true of the n-d cubes too.
Yeah, if you’re small enough compared to the surface, they’re both flat. In the intermediate regime where r < R but not r << R, the second-order effects of the sphere matter and give you the needle effect whereas the cube pushes all those effects into the edges and corners, so the faces still look flat at intermediate scales.
I think this is another way of saying that almost all of the volume of a high-dimensional sphere is at the outer surface.
No?
It’s a statement about your small neighborhood when you at the surface of the n-d sphere. And in low n it’s like half inside the sphere. In high n it’s more like a needle touching your position.
I’m not sure how that ^follows from the fact that most of volume of a high-dimensional sphere is at the outer surface.
The needle effect occurs because there’s more volume just above the surface than just below it.
EDIT: If you imagine expanding the large sphere slightly to encompass the entire small sphere, most of the volume of the new large sphere will be outside of the original large sphere.
Well, if you are on the face on n-d cube, half of your neighborhood would be inside, no matter what n. But “most volume in the peel” is true of the n-d cubes too.
Yeah, if you’re small enough compared to the surface, they’re both flat. In the intermediate regime where r < R but not r << R, the second-order effects of the sphere matter and give you the needle effect whereas the cube pushes all those effects into the edges and corners, so the faces still look flat at intermediate scales.