Symmetrization inequalities and optimal transport methods  
are used in the proof of many geometric-functional inequalities in  
sharp form. Sharp quantitative refinements of several isoperimetric  
principles, including the Euclidean and Gaussian isoperimetric  
inequalities, as well as the Wulff theorem and the Brunn-Minkowski  
inequality on convex sets, can be attacked on starting from these  
methods.