Polar Duality and the Donoho--Stark Uncertainty Principle

Abstract: Polar duality is a fundamental geometric concept that can be interpreted as a form of Fourier transform between convex sets. Meanwhile, the Donoho-Stark uncertainty principle in harmonic analysis provides a framework for comparing the relative concentrations of a function and its Fourier transform. Combining the Blaschke--Santal\'o inequality from convex geometry with the Donoho--Stark principle, we establish estimates for the trade-off of concentration between a square integrable function in a symmetric convex body and that of its Fourier transform in the polar dual of that body. In passing, we use the Donoho-Stark uncertainty principle to establish a new concentration result for the Wigner function.