In the context of the Wigner-Weyl phase space formulation of quantum mechanics, a version of the uncertainty relations invariant under affine canonical transformations is derived. For a fixed Wigner distribution function possessing a finite covariance, "directions" of minimal uncertainty are found. The geometry of the Wigner ellipsoid and its Legendre transform, the dual Wigner ellipsoid, both of which are associated with the same covariance, is discussed. The results obtained are generalizations of the well-known fact that, for one degree of freedom, the area of the Wigner ellipse must be of order or larger. Instead of area, which is an invariant only when n = 1, these results involve Poincar invariants of certain curves and surfaces. 1990 American Institute of Physics.
