Transport-majorization to analytic and geometric inequalities

We introduce a transport-majorization argument that establishes majorization in the convex order between two densities, based on control of gradient transportation map them. As applications, we give elementary derivations some delicate Fourier analytic inequalities, which turn yield geometric “slicing-inequalities” both continuous and discrete settings. further consequence our investigation prove any strongly log-concave probability density majorizes Gaussian thus maximizes Rényi Tsallis entropies all orders among densities.