On 20/05/2020 21:07, Dan Asimov wrote:
Let the Hilbert (regular) simplex Δ be defined as the closed convex hull of that orthonormal basis:
Δ = CC({e_n}).
This is by analogy with the n-dimensional regular simplex Δ_n (n >= 0) in H, defined as the closed convex hull of the basis of R^(n+1) in H:
CC({e_1,...,e_n})
You want that to go up to e_{n+1}, right?
Now suppose we want to walk from the centroid e_1 of Δ_0 to the centroid of Δ_1 to the centroid of Δ_2 to ... to the centroid of Δ_n to ..., hopefully ending up at the centroid of Δ.
1) Assuming these steps for n = 1, 2, 3, ... are taken in straight segments, what is the net distance traveled in the limit?
The centroid of Δ, to whatever extent that's well defined, is at zero. The centroid of Δ_0 is e_1. The distance is 1.
2) What is the total distance traveled, in the limit?
The centroid of Δ_n is 1/n+1 (1,...,1,0,...) where there are n+1 1s. So the vector from the centroid of Δ_n-1 to that of Δ_n has n coordinates that are -1/n(n+1), and one that's +1/(n+1). So the distance is the square root of n/n^2(n+1)^2 + 1/(n+1)^2, or 1 / sqrt(n(n+1)). For large n this looks like 1/n so this sum doesn't converge: the distance is infinite.
3) If all these centroids were projected radially onto the unit sphere S of H, obtaining points d_0, d_1, d_2, ..., d_n, ... on S, what would be the net distance traveled from d_0 to d_1, then from d_1 to d_2, etc., in the limit? Again assume each step is in a straight line.
After projection to the unit sphere, the centroid of Δ_n becomes 1/sqrt(n+1) (1,...,1,0,...) where there are n+1 1s. So the difference vector has n coordinates that are 1/sqrt(n+1)-1/sqrt(n) and one that's 1/sqrt(n+1). In the unlikely event that I've done the algebraic manipulations right, that means that the distance squared is 2(1 - sqrt(n/(n+1))). This is of order 1/n, so its square root is of order 1/sqrt(n), so the total distance is even more infinite than in part 2. (It seems like we're supposed to find something here terribly surprising. I personally don't have strong enough intuitions about this stuff for that to be so.) -- g