Balls have the worst best Sobolev inequalities

Using transportation techniques in the spirit of Cordero-Erausquin, Nazaret and Villani [A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities. Adv. Math. 182(2), 307-332, (2004)], we establish an optimal non parametric trace Sobolev inequality, for arbitrary locally Lipschitz domains in R^n. We deduce a sharp variant of the Brézis-Lieb trace Sobolev inequality [Brézis, H. and Lieb, E. Sobolev inequalities with a remainder term. J. Funct. Anal. 62, 73-86, (1985)], 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 [op cit].