Letter to the EditorTwo geometric interpretations of the multidimensional Hardy uncertainty principle

- We state a multidimensional version of Hardy's uncertainty principle.
- The Hardy uncertainty principle is equivalent to a statement about the symplectic capacity of the Hardy ellipsoid.
- We express this result in terms of the projections of the Hardy ellipsoid on the x- and p-spaces.

Hardy's uncertainty principle says that a square integrable function and its Fourier transform cannot be simultaneously arbitrarily sharply localized. We show that a multidimensional version of this uncertainty principle can be best understood in geometrical terms using the fruitful notion of symplectic capacity, which was introduced in the mid-eighties following unexpected advances in symplectic topology (Gromov's non-squeezing theorem). In this geometric formulation, the notion of Fourier transform is replaced with that of polar duality, well-known from convex geometry.