01/31/2008, 11:13 AM

Well when I read the defintion of hypersurface in Wikipedia:

In algebraic geometry, a hypersurface in projective space of dimension n is an algebraic set that is purely of dimension n − 1. It is then defined by a single equation F = 0, a homogeneous polynomial in the homogeneous coordinates. (It may have singularities, so not in fact be a submanifold in the strict sense.)

Letting out the subtleties, the hypersurface of hypervolume -I must be dI= - I lnI = -pi/2

And, for hypervolume I -> dI = IlnI = pi/2.

Because there is nothing else given, the only way to reduce infinite dimensionality of hypervolume I to (infinity -1 ) is by taking a differential of it.

In algebraic geometry, a hypersurface in projective space of dimension n is an algebraic set that is purely of dimension n − 1. It is then defined by a single equation F = 0, a homogeneous polynomial in the homogeneous coordinates. (It may have singularities, so not in fact be a submanifold in the strict sense.)

Letting out the subtleties, the hypersurface of hypervolume -I must be dI= - I lnI = -pi/2

And, for hypervolume I -> dI = IlnI = pi/2.

Because there is nothing else given, the only way to reduce infinite dimensionality of hypervolume I to (infinity -1 ) is by taking a differential of it.