  We approach uncertainty principles of Cowling-Price-Heis-\\enberg-type as a
variational principle on modulation spaces. In our discussion we are naturally
led to compact localization operators with symbols in modulation spaces. The
optimal constant in these uncertainty principles is the smallest eigenvalue of
the inverse of a compact localization operator. The Euler-Lagrange equations
for the associated functional provide equations for the eigenfunctions of the
smallest eigenvalue of these compact localization operators. As a by-product of
our proofs we derive a generalization to mixed-norm spaces of an inequality for
Wigner and Ambiguity functions due do Lieb.
