The sharp Sobolev and isoperimetric inequalities split twice

By Jie Xiao. 2007

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Sharp convex Lorentz–Sobolev inequalities (2011, unknown): “For strengthened versions of (1), see, e.g., [7,14,17,20,30,46,75,77] .”
An asymmetric affine Pólya–Szegö principle (2012, unknown): “, [3,8,20,25,26,42,58,67,71]), and the references therein).”
Valuations on Sobolev spaces (2012, background): “Moreover, the optimal Sobolev body has proved to be critical in recent results on affine isoperimetric inequalities (see [13, 24, 40, 44, 45, 59, 60]).”
Optimal Sobolev norms and the Lp Minkowski problem (2006, background): “Zhang [34] and the authors [22] have recently established a sharp Lp affine Sobolev inequality (a version of the L1 affine Sobolev inequality involving capacity has recently been established by Xiao [33]).”
Asymmetric affine Lp Sobolev inequalities (2009, background): “In recent years, many variations and generalizations have been obtained, see, e.g., [2, 3, 6, 8, 11, 32, 33, 37 ] and the references therein.”