# What happens to the volume of an ellipsoid as the number of dimensions increases? [closed]

Would the volume of an ellipsoid continuously increase if one keeps adding radii along new dimensions? What is the volume of ellipsoid with infinite dimensions?

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## closed as too localized by Pete L. Clark, Deane Yang, Bill Johnson, Ryan Budney, José Figueroa-O'FarrillSep 24 '10 at 1:29

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The $n$-dimensional volume goes to zero rather quickly as $n$ increases, as you can read about at en.wikipedia.org/wiki/N-sphere. Specifically, a sphere of radius 1 in $\mathbb{E}^n$ has volume $\pi^{n/2}/\Gamma(n/2+1)$. For intuition, note that in 1 dimension, the (unit) 1-sphere and the 1-cube both have volume 2. In 2 dimensions, the 2-sphere has volume \pi and the 2-cube has volume 4. Far from being distance 1 from the origin, the corners of the unit $n$-cube are distance $\sqrt{n}$ from the origin... they get farther and farther away in the Euclidean norm. – Eric Tressler Sep 23 '10 at 22:36
To add to Eric's comment, since the question is about ellipsoids, I'd just say that the sphere of radius R in $\mathbb{R}^n$ has volume $\pi^{n/2}R^n/\Gamma(n/2 + 1)$. Ellipsoids have lots of different radii as it were and so instead of having $R^n$ you'd have a product of the different radii. So the volume would behave differently according to the behaviour of your sequence of new radii if that makes any sense. Perhaps en.wikipedia.org/wiki/… might help with the second question. – Spencer Sep 23 '10 at 22:52