What is the fraction of volume of unit hypersphere centered at one of the vertices of hypercube to that of hypercube?

Question

consider a hyper-cube of n-dimension having a length of "r" units across each dimension. If a unit n-dimensional sphere is present at one of the vertices of the hyper-cube. what fraction of volume of the hyper-cube is occupied by hyper-sphere?

Answer 1

There are some simple cases.

For $r>1$, the hypercube is big enough that the hypersphere will only intersect half its faces. The hypercube will cut out $2^{-n}$ of the hypersphere. So in the plane, you'd cut out a quadrter of the circle, in 3d you'd get one quarter of a sphere, and so on. Combine that with the volume of the $n$-ball and you get

$$2^{-n}V_n = \frac{\pi^{n/2}}{2^n\,\Gamma(\frac n2+1)}$$

The other extreme case is $r\sqrt n<1$ resp. $r<n^{-1/2}$. In this case, the hypercube is completely contained within the hypersphere, so the volume is simply that of the cube, $r^n$.

In between these extremal situations, there are a bunch of other cases. Whenever $r\sqrt k=1$ for some $k\in\{1,2,\dots,n\}$, the hypersphere will pass through some corners of the hypercube. For every range between (and including) such special values, you have good chances of finding a closed form expression describing the volume within that interval. At least if you do this for some specific $n$. It would be quite some work, and I don't claim to answer that aspect of your question. The occurrence of the gamma function in the equation above already hints at the notation you'd likely have to deal with if you attempt this for arbitrary dimension $n$.