Hi, OP! That's a good problem. Have you looked at the theorem about the spaces within a grid of N-spheres? As you increase dimensions, at a certain point, the N-spheres you can fit inside the gaps within a grid of hyperspheres actually become larger than the grid itself.
It's a really interesting result and a great one to wrestle with when you start trying to reason about higher-dimensional spaces.
https://math.stackexchange.com/questions/2930015/spheres-cause-contradictions-in-dimensions-10-and-moreMy intuition about the hypersphere volume reducing to zero would be an inverse of the above principle; as the dimensions increase to infinity, the 'space' between the spheres becomes overwhelmingly large and the sphere volumes are effectively reduced to 0 at the limit. (If you want a proof on why that decreasing proportion doesn't converge to some x>0, you'll have to ask someone else! It's fairly intuitive, though.)
Hope that helps.
