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arXiv · 2110.03641

Transport-majorization to analytic and geometric inequalities

James Melbourne·Cyril Roberto
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Abstract

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

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James Melbourne, Cyril Roberto. 2021-10-07. Transport-majorization to analytic and geometric inequalities. https://doi.org/10.1016/j.jfa.2022.109717

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