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We define and study the notion of quantum polarity, which is a kind ofgeometric Fourier transform between sets of positions and sets of momenta.Extending previous work of ours, we show that the orthogonal projections of thecovariance ellipsoid of a quantum state on the configuration and momentumspaces form what we call a dual quantum pair. We thereafter show that quantumpolarity allows solving the Pauli reconstruction problem for Gaussianwavefunctions. The notion of quantum polarity exhibits a strong interplaybetween the uncertainty principle and symplectic and convex geometry and ourapproach could therefore pave the way for a geometric and topological versionof quantum indeterminacy. We relate our results to the Blaschke-Santal\'oinequality and to the Mahler conjecture. We also discuss the Hardy uncertaintyprinciple and the less-known Donoho--Stark principle from the point of view ofquantum polarity.

Quantum Polar Duality and the Symplectic Camel: a Geometric Approach to Quantization