Affine Moser-Trudinger and Morrey-Sobolev inequalities (Q1047926)

With the standard \(\mathcal L^p\) energy of gradient replaced by an affine energy, by utilizing the rapidly evolving \(\mathcal L^p\) Brunn-Minkowski theory of Convex Geometry, in particular a recently established affine isoperimetric inequality for convex bodies, an affine version of the euclidean Moser-Trudinger inequality is established. By simple comparison of the affine energy and the \(\mathcal L^p\) energy of gradient, it is noted that this new affine version of Moser-Trudinger inequality is stronger than the euclidean one.