In high dimension, volume of unit ball goes to 0. It is definitely less than volume of ball in three dimension.
The question is, then, how the d-dimension ball can contain three dimension ball?
I thought that 1 volume in three dimension is considered by volume of unit cube.
And 1 volume in d-dimension is considered by volume of unit cube in d-dimension.
So I think that the way to calculate volume in different dimension is different.
Is this right answer? I don't know how to explain this.
Thanks in advance.
A good way to think about this is in $3$ vs $2$ dimensions because we can visualize what is going on. The volume of a radius $\frac 12 \ 3-$ball is $\frac 43\pi \frac 18=\frac \pi 6$. The area of a radius $\frac 12\ 2-$ball (a disk) is $\pi \frac 14=\frac \pi 4$, which is greater than the volume of the sphere that contains it. The volume of the disk is $0$ because its extent in the third direction is zero. There is no contradiction because we are talking about different things.