  We prove a new type of Poincar\'e inequality on abstract Wiener spaces for a
family of probability measures which are absolutely continuous with respect to
the reference Gaussian measure. This class of probability measures is
characterized by the strong positivity (a notion introduced by Nualart and
Zakai in [17]) of their Radon-Nikodym densities. Measures of this type do not
belong in general to the class of log-concave measures, which are a wide class
of measures satisfying the Poincar\'e inequality (Brascamp and Lieb [2]). Our
approach is based on a point-wise identity relating Wick and ordinary products
and on the notion of strong positivity which is connected to the non negativity
of Wick powers. Our technique leads also to a partial generalization of the
Houdr\'e and Kagan [8] and Houdr?\'e and P?\'erez-Abreu [9] Poincar\'e-type
inequalities.
