Sharp Affine L P Sobolev Inequalities

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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]).”
The even Orlicz-Minkowski problem (2010, methodology): “A byproduct is a new approach to the solution of the classical Minkowski problem.”
Functional affine-isoperimetry and an inverse logarithmic Sobolev inequality (2011, background): “There is a vast amount of literature on logarithmic Sobolev inequalities and related topics, e.g. [2, 5, 6, 9, 11, 15, 21].”
R\'enyi Divergence and $L_p$-affine surface area for convex bodies (2011, background): “Introduction There exists a fascinating connection between convex geometric analysis and information theory.”
An asymmetric affine Pólya–Szegö principle (2012, methodology): “It was shown in [56] that Ep( f ) ≤ ‖∇ f ‖p, (4)”