>Alternatively almost all volume of n-sphere is close to the surface. How does this go against intuition? Intuition from 1/2/3d tells me that the volume of an

>Alternatively almost all volume of n-sphere is close to the surface. How does this go against intuition? Intuition from 1/2/3d tells me that the volume of an N-ball is O(r^N), and indeed it is the case in higher dimensions. Therefore it’s easy to see that the difference between the volume of an N-ball of radius r and an N-ball of radius (r + epsilon) will grow exponentially with N.