Is there some obvious way (that I seem to be missing) to see that the centrally inscribed D-sphere must touch all of the other spheres in high dimensions?
That's probably a stupid question, but while that fact is intuitively obvious for D={2,3} -- as this problem tries to demonstrate -- higher dimensions are unintuitively WEIRD.
You can argue by symmetry that if the central sphere touches one of the corner spheres, it must touch them all. And it must touch one because otherwise you would increase it's radius until it did.
Well we can calculate the touch point for one sphere, and we know that it would overlap that sphere if it was radius sqrt(D).
And all the other spheres are simple symmetrical mirrors, so how could it not touch all of them at the same time it touches one? That should scale to an arbitrary number of dimensions, right?
I also found it very weird, but here's my intuition.
There are 2^k > 512 spheres stuck to eachother across k-d (pretend k=9). The line from the center to the point where the inner sphere touches one of outer spheres has to shortcut through all k dimensions to get from the center to the sphere.
This distance has been massively inflated due to the number of dimensions. But the distance to the edge of the box hasn't been inflated - it's just constant, so the inner sphere breaks out.
OK, Another reference is [1], that agrees with the result given by the OP.
I'm not trying to do research here, I'm just boggling at the unintuitive result, and trying to see if there might be a flaw in the chain of logic. The fact that this is "well known" is enough to scare me off from barking up this particular tree.
That's probably a stupid question, but while that fact is intuitively obvious for D={2,3} -- as this problem tries to demonstrate -- higher dimensions are unintuitively WEIRD.