  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 cases of
equality in this case. For this new inequality, no Euclidean geometric
structure is needed.
