# 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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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
–  S. Carnahan Sep 24 '10 at 2:50
There is something in this question that hasn't been asked before, namely about ellipsoids in infinite dimensions. The answer, though, is that it is very hard to come up with good theories of "volume" in infinite dimensions, for a number of interesting and related reasons. –  Theo Johnson-Freyd Sep 24 '10 at 3:12