  We define and study the notion of quantum polarity, which is a kind of
geometric Fourier transform between sets of positions and sets of momenta.
Extending previous work of ours, we show that the orthogonal projections of the
covariance ellipsoid of a quantum state on the configuration and momentum
spaces form what we call a dual quantum pair. We thereafter show that quantum
polarity allows solving the Pauli reconstruction problem for Gaussian
wavefunctions. The notion of quantum polarity exhibits a strong interplay
between the uncertainty principle and symplectic and convex geometry and our
approach could therefore pave the way for a geometric and topological version
of quantum indeterminacy. We relate our results to the Blaschke-Santal\'o
inequality and to the Mahler conjecture. We also discuss the Hardy uncertainty
principle and the less-known Donoho--Stark principle from the point of view of
quantum polarity.
