A two-dimensional n-ball is a circle. If you extend it into 3 dimensions, by default it is a cylinder, of h=1 and r=1. That’s not a 3-dimensional n-ball however.
If you increase the dimensions of that 2-dimensional n-ball into 3 dimensions, it becomes a 3-dimensional n-ball—-a sphere of r=1.
And now you have a sphere that is smaller than the circle that was extended into 3 dimensions as a cylinder. The ball-ness of the n-ball formula chopped off the corners of the cylinder.
And that chopping off continues in higher dimensions.
For r=1
A two-dimensional n-ball is a circle. If you extend it into 3 dimensions, by default it is a cylinder, of h=1 and r=1. That’s not a 3-dimensional n-ball however.
If you increase the dimensions of that 2-dimensional n-ball into 3 dimensions, it becomes a 3-dimensional n-ball—-a sphere of r=1.
And now you have a sphere that is smaller than the circle that was extended into 3 dimensions as a cylinder. The ball-ness of the n-ball formula chopped off the corners of the cylinder.
And that chopping off continues in higher dimensions.