Polar Duality and the Donoho--Stark Uncertainty Principle
Maurice de Gosson
TL;DR
The paper integrates polar duality from convex geometry with the Donoho–Stark uncertainty principle to quantify how sharply a function and its Fourier transform can be simultaneously localized within a symmetric convex body and its polar dual. By leveraging Blaschke–Santaló inequalities, it derives precise concentration bounds and a Mahler-volume framework that links geometry to uncertainty, including a concentration result for the Wigner function. The work connects quantum blobs and symplectic geometry to fundamental limits of localization, offering a geometric lens on quantum indeterminacy and providing quantitative trade-offs that extend beyond Gaussian or variance-based descriptions. These results illuminate how convex-geometric structure governs phase-space concentration and yield new insights into concentration phenomena in the Wigner representation.
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--Santaló 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.