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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1$\begingroup$ 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. $\endgroup$– Eric TresslerCommented Sep 23, 2010 at 22:36
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1$\begingroup$ 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. $\endgroup$– SpencerCommented Sep 23, 2010 at 22:52
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$\begingroup$ See also: mathoverflow.net/questions/6160/… $\endgroup$– S. Carnahan ♦Commented Sep 24, 2010 at 2:48
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$\begingroup$ Also: mathoverflow.net/questions/8258/… $\endgroup$– S. Carnahan ♦Commented Sep 24, 2010 at 2:50
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$\begingroup$ 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. $\endgroup$– Theo Johnson-FreydCommented Sep 24, 2010 at 3:12
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Note that it is enough to consider the case of the unit sphere, since to get an ellipsoid from the unit sphere you just have to apply a diagonal matrix whose entries are the semi axis lengths, which multiplies the volume by the determinant, I.e., the product of the semi axis lengths. Also,it is easy to get the volume of the n-sphere inductively, with just one easy integration by slicing it by planes parallel to an axis and noting that the slices are spheres of one lower dimension and easily computed radii.
answered Sep 24, 2010 at 0:56