Balls have the worst best Sobolev inequalities

Using transportation techniques in the spirit of Cordero-Erausquin, Nazaret and Villani 7], we establish an optimal non parametric trace Sobolev inequality, for arbitrary locally Lipschitz domains in ℝⁿ. We deduce a sharp variant of the Brézis-Lieb trace Sobolev inequality 4], containing both the isoperimetric inequality and the sharp Euclidean Sobolev embedding as particular cases. This inequality is optimal for a ball, and can be improved for any other bounded, Lipschitz, connected domain. We also derive a strengthening of the Brézis-Lieb inequality, suggested and left as an open problem in 4]. Many variants will be investigated in a companion article 10].