  We use the notion of polar duality from convex geometry and the theory of
Lagrangian planes from symplectic geometry to construct a fiber bundle over
ellipsoids that can be viewed as a quantum-mechanical substitute for the
classical symplectic phase space. The total space of this fiber bundle consists
of geometric quantum states, products of convex bodies carried by Lagrangian
planes by their polar duals with respect to a second transversal Lagrangian
plane.. Using the theory of the John ellipsoid we relate these geometric
quantum states to the notion of "quantum blobs" introduced in previous work;
quantum blobs are the smallest symplectic invariant regions of the phase space
compatible with the uncertainty principle. We show that the set of equivalence
classes of unitarily related geometric quantum states is in a one-to-one
correspondence with the set of all Gaussian wavepackets.
