I have recently been shown the gamma function and a few of its uses, and one of those is calculating the measure of the unit ball in $\Bbb{R}^n$. The formula shows the measure going to zero (rather fast, actually) as n grows greater.
I have understood the proof, but it was achieved through some artifacts of integration, and it does not provide a geometric understanding of what is happening.
The result was justified by mentioning that the (hyper)cube that surrounds the sphere has a diagonal of $2^{n/2}$ and that since the measure is, intuitively, how many tiny cubes can I fit inside the sphere, if the volume of the cube is fixed and grows bigger relative to the sphere, then it is the sphere that is becoming smaller overall.
I cannot however visualize this: in fact, the cube has a side length of 2, which translates to a volume of 2^n, which is also growing...
Moreover, an $\infty$-dimensional space has a unit ball of measure $\infty$ (I proved this for $C(0, 1)$), which seems to contrast with the "limit" of the size of $D^n$, the ball of $\Bbb{R}^n$.
Can anyone give an intuitive ("geometric") reason for why the unit balls grow smaller as n increases?
