A sharp affine Lp Sobolev inequality for functions on Euclidean n-space is established. This new inequality is significantly stronger than (and directly implies) the classical sharp Lp Sobolev inequality of Aubin and Talenti, even though it uses only the vector space structure and standard Lebesgue measure on ℝn. For the new inequality, no inner product, norm, or conformal structure is needed; the inequality is invariant under all affine transformations of ℝn.
"Sharp Affine LP Sobolev Inequalities." J. Differential Geom. 62 (1) 17 - 38, September, 2002. https://doi.org/10.4310/jdg/1090425527