The sharp affine \(L^2\) Sobolev trace inequality and variants (Q681625)

The authors prove a significantly stronger sharp affine \(L^2\) Sobolev trace inequality on \({\mathbb R}^n_+\). A remarkable and surprising feature of the new inequality is its independence of the norm chosen for the Euclidean space. The proof requires the \(L^p\) Busemann-Petty centroid inequality, and the method reveals in an explicit and elementary way the geometric nature behind affine inequalities.