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How can one find a general formula to find what fraction of a $n$-dimensional sphere is the volume of the inscribed cube?

Context: the problem emerged out of curiosity starting from the $3$-D case, and I would like to have some hints on the kind of approach that may be useful.

Dal
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    There are explicit formulae for the volume of each. Given that the long diagonal of the cube will be equal to the diameter of the sphere, it is not hard to find their ratio. – Dan Rust Sep 29 '14 at 22:30
  • Thank you. I would like to know how to find out these formulae (and consequently the formula for the ratio) without looking them up. – Dal Sep 29 '14 at 22:32
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    Have you tried a cursory google search? I'm pretty search 'volume of an n-dimensional sphere' will offer many texts on the topic. – Dan Rust Sep 29 '14 at 22:34
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    Wikipedia appears to offer 4 proofs, ranging from recurrence relations to basic multiple integrals, and Gaussian integrals. – Dan Rust Sep 29 '14 at 22:38

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The volume of a n-sphere with radius $\displaystyle R$ is $\displaystyle \frac{\pi^{\frac{n}{2}}}{\Gamma{(\frac{n}{2}+1)}} R^n$. The volume of a n-cube with side $\displaystyle S$ is $\displaystyle S^n$, and its longest diagonal is $\displaystyle S \sqrt{n}$. In this case we have $S \sqrt{n}=2R$ and then $\displaystyle S=\frac{2R}{\sqrt{n}}$, which leads to a n-cube volume equal to $\displaystyle \frac{(2R)^n}{n^{\frac{n}{2}}}$.

The ratio of the inscribed n-cube volume to the total n-sphere volume is then

$$\displaystyle \frac{(2R)^n \, \Gamma{(\frac{n}{2}+1)} }{n^{\frac{n}{2}} \pi^{\frac{n}{2}} R^n}$$

which can be simplified in

$$\displaystyle \frac{2^n \, \Gamma{(\frac{n}{2}+1)} }{(n\pi)^{\frac{n}{2}}}$$

The first values for $n=2,3,4,5...$ are $\displaystyle \frac{2}{\pi}$, $\displaystyle\frac{2}{\sqrt{3}\pi}$,$\displaystyle\frac{2}{\pi^2}$, $\displaystyle\frac{12}{5 \sqrt{5} \pi^2}$, and so on. Note that, as $n \rightarrow \infty$, the ratio converges to zero.

Anatoly
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  • Can we say that longest diagonal equals the 2R? As I try to visualize this in 3D, I can see how one side of the cube equals 2R, but the longest diagonal cuts through the sphere and extends beyond it. – Jxson99 Sep 07 '22 at 18:50
  • Note that the cube is inscribed in the sphere. In this case, its major diagonal is equal to the diameter. The side of the cube would be equal to the diameter (and the diagonal would be greater than the diameter) if the cube were circumscribed, but this is not the case. – Anatoly Sep 08 '22 at 09:55
  • You're right - it's the cube that's inscribed in the sphere, not the other way around. Ty. – Jxson99 Sep 08 '22 at 15:16