A note on some Poincar\'e inequalities on convex sets by Optimal   Transport methods
  We show that a class of Poincar\'e-Wirtinger inequalities on bounded convex sets can be obtained by means of the dynamical formulation of Optimal Transport. This is a consequence of a more general result valid for convex sets, possibly unbounded. 
Lorenzo Brasco, Filippo Santambrogio
math.AP math.FA