Poincar{é} inequalities and integrated curvature-dimension criterion for generalised Cauchy and convex measures

We obtain new sharp weighted Poincar{é} inequalities on Riemannian manifolds for a general class of measures. When specialised to generalised Cauchy measures, this gives a unified and simple proof of the weighted Poincar{é} inequality for the whole range of parameters, with the optimal spectral gap, the error term and the extremal functions.