Why sqrt(2)? Well, if you take two samples from an n-dimensional Gaussian, it’s easy to show that the inner product of the two vectors is likely to be small. So the inner product of two random unit vectors chosen from S^(n-1) should be small too; so the angle between them should be close to 90 degrees; and Pythagoras does the rest. Or, we can appeal to Elk’s Theorem: the n-dimensional hypersphere, like the brontosaurus, is thin at one end, much MUCH thicker in the middle, and thin again at the far end. Jim Propp On Tuesday, May 29, 2018, James Propp <jamespropp@gmail.com> wrote:
I’m betting on sqrt(2).
Jim Propp
On Tuesday, May 29, 2018, Dan Asimov <dasimov@earthlink.net> wrote:
Let D(n) denote the average distance through R^(n+1) between two points uniformly distributed on the unit n-sphere S^n.
E.g., D(1) = 4/π, D(2) = 4/3.
Puzzle: Find the limit
lim D(n) n—>oo
—Dan
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