> volume of an n+1-ball can be of a smaller volume than an n-ball
Volume is the ratio of the n-ball to the n-cube.
Saying that n+1-ball has “smaller volume” than the n-ball is equivalent to saying that the ratio of the n+1-ball to the n+1-cube is smaller than the ratio of the n-ball to the n-cube.
> I understand how the proportional space of an n-ball bounded in an n-cube can go to zero
The volume comment is saying that the ratio of the n-ball to its bounding n-cube goes to zero faster than (base-2) exponential:
Volume is the ratio of the n-ball to the n-cube.
Saying that n+1-ball has “smaller volume” than the n-ball is equivalent to saying that the ratio of the n+1-ball to the n+1-cube is smaller than the ratio of the n-ball to the n-cube.
> I understand how the proportional space of an n-ball bounded in an n-cube can go to zero
The volume comment is saying that the ratio of the n-ball to its bounding n-cube goes to zero faster than (base-2) exponential:
The bounding n-cube is 2^n times the unit n-cube.