  The notion of polarity between sets, well-known from convex geometry, is a
geometric version of the Fourier transform. We exploit this analogy to propose
a new simple definition of quantum indeterminacy, using what we call
"hbar-polar quantum pairs", which can be viewed as pairs of position-momentum
indeterminacy with minimum spread. The existence of such pairs is guaranteed by
the usual uncertainty principle, but is at the same time more general. We use
recent advances in symplectic topology to show that this quantum indeterminacy
can be measured using a particular symplectic capacity related to action and
which reduces to area in the case of one degree of freedom. We show in addition
that polar quantum pairs are closely related to Hardy's uncertainty principle
about the localization of a function and its Fourier transform.
