The sharp affine $$L^2$$ Sobolev trace inequality and variants

We establish a sharp affine \(L^p\) Sobolev trace inequality by using the \(L_p\) Busemann–Petty centroid inequality. For \(p = 2\), our affine version is stronger than the famous sharp \(L^2\) Sobolev trace inequality proved independently by Escobar and Beckner. Our approach allows also to characterize all extremizers in this case. For this new inequality, no Euclidean geometric structure is needed.