Shift inequalities of Gaussian type and norms of barycentres

We derive the equivalence of different forms of Gaussian type shift inequal ities. This completes previous results by Bobkov. Our argument strongly rel ies on the Gaussian model for which we give a geometric approach in terms o f norms of barycentres. Similar inequalities hold in the discrete setting, they improve the known results on the so-called isodiametral problem for th e discrete cube. The study of norms of barycentres for subsets of convex bo dies completes the exposition.