I am trying to prove the curse of dimensionality and on Wikipedia https://en.wikipedia.org/wiki/Curse_of_dimensionality The volume of Hypersphere is given as $\frac{2r^d \pi^{d/2}}{d\;\Gamma (d/2)}$ Can anybody help me that how we can derive this formula? as far as I know, the volume for $N$-dimensional hypersphere is $\frac{r^n\pi^{n/2}}{\;\Gamma (n/2+1)}$

  • $\begingroup$ You only need the $r^n$ term to demonstrate the curse of dimensionality $\endgroup$ – Henry Mar 24 '20 at 17:33
  • $\begingroup$ In most texts I've seen, your formula is for the measure of the $n$-ball, not the $n$- or $n+1$-sphere. (In other words, the sphere is the surface, not the bulk.) $\endgroup$ – J.G. Mar 24 '20 at 17:34
  • $\begingroup$ @Henry can you show me how? $\endgroup$ – christopher Mar 24 '20 at 17:39
  • $\begingroup$ $99\%$ or the volume of a hyperball radius is between $\sqrt[n]{0.01}r$ and $r$ from the origin. For large $n$ you find $\sqrt[n]{0.01}r$ is close to $r$; e.g. for $n=10000$ it is about $0.99954r$ $\endgroup$ – Henry Mar 24 '20 at 17:45

The Gamma function has the property $\Gamma(z+1)=z\Gamma(z)$, so $$ \frac {r^{n}\pi ^{{n/2}}}{\Gamma (n/2+1)} = \frac {r^{n}\pi ^{{n/2}}}{\frac n2\Gamma (n/2)} = \frac {2r^{n}\pi ^{{n/2}}}{n\Gamma (n/2)}. $$ The two formulas you have therefore agree.

For more details on finding the volumes of balls and spheres, the Wikipedia links should get you started.

  • $\begingroup$ very nice explanation and the comment above mentioned as well that n-balls cover the area till n space whereas n-sphere covers the area till n+1 space. I was making the mistake of considering both the same. $\endgroup$ – christopher Mar 24 '20 at 17:49
  • $\begingroup$ I once derived the volume of a sphere by using an integral of the cross-section circles making thin disks of $\delta x$ thickness. Then I did the same thing using a cross-section of a 3D sphere instead to get the volume of a 4D sphere. Is that the same thing your answer shows? $\endgroup$ – Nεo Pλατo Mar 24 '20 at 18:45
  • $\begingroup$ @Plato There's a number of ways to find the volumes of spheres and balls. I didn't really use any of them; I just pointed out that the two formulas are the same and gave links for further information. If you want to compare two methods or have any other questions, please consider asking a separate question. $\endgroup$ – Joonas Ilmavirta Mar 24 '20 at 18:48
  • $\begingroup$ Nevermind. I thought you were explaining how the formula came about. The consequences of perusing through text $\endgroup$ – Nεo Pλατo Mar 24 '20 at 18:50

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