We show that the covariance matrix of a quantum state can be reconstructed from position measurements using the simple notion of polar duality, familiar from convex geometry. In particular, all multidimensional Gaussian states (pure or mixed) can in principle be reconstructed if the quantum system is well localized in configuration space. The main observation which makes this possible is that the John ellipsoid of the Cartesian product of the position localization by its polar dual contains a quantum blob, and can therefore be identified with the covariance ellipsoid of a quantum state.