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. By combining the Blaschke--Santaló inequality from convex geometry with the Donoho--Stark principle, we establish sharp estimates that express the quantum mechanical uncertainty principle in terms of polar duality. Our central result shows that the sum of the probabilities of position concentration near a convex body and momentum concentration near its polar dual approaches one, with an error term that vanishes rapidly as the number of degrees of freedom increases.