Let H be the Hilbert space of square-summable sequences of reals:
H = 𝓵^2(Z+; R).
Then the vectors e_n, n in Z+, that are 1 in the nth place and 0 elsewhere form an orthonormal Hilbert basis.
For any subset X of H, the closed convex hull CC(X) of X is defined as the intersection of all closed half-spaces L^(-1)([c,oo)) that contain X, where L is a continuous nonzero linear map L : H —> R and c is a real number.
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})
(though of course the convex hull of a finite set is automatically closed).
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?
2) What is the total distance traveled, in the limit?
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.
—Dan
