Affine Moser-Trudinger and Morrey-Sobolev inequalities

An affine Moser–Trudinger inequality, which is stronger than the Euclidean Moser–Trudinger inequality, is established. In this new affine analytic inequality an affine energy of the gradient replaces the standard Ln energy of gradient.The geometric inequality at the core of the affine Moser–Trudinger inequality is a recently established affine isoperimetric inequality for convex bodies. Critical use is made of the solution to a normalized version of the L^n Minkowski Problem. An affine Morrey–Sobolev inequality is also established, where the standard L p energy, with p > n, is replaced with the affine energy