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