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 L-n 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 by the affine energy.