[math-fun] Hilbert space puzzle
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
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
On Wed, May 20, 2020 at 4:07 PM Dan Asimov <dasimov@earthlink.net> wrote:
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.
I don't think this is the definition you want. This gives the convex hull of a single point P as the ray from P away from the origin, and the convex hull of two points P and Q as the part of the plane containing the origin, P, and Q bounded by the line segment through O and P and the rays from P and Q pointing away from the origin. I think you either want to take the intersection not only of sets of the form L^-1([c, inf)) but also sets of the form L^-1([b, c]), or at least sets of the form L^-1((-inf, c]). Alternatively, you could use your definition, but extend it to include continuous affine functions L (the sum of a continuous linear functional and a constant). Andy
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
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participants (3)
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Andy Latto -
Dan Asimov -
Gareth McCaughan