Transport-majorization to analytic and geometric inequalities |
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Institution: | 1. Centro de Investigación en Matemáticas, Probabilidad y Estadísticas.: 36023 Guanajuato, Gto, Mexico;2. Université Paris Nanterre, Modal''X, UMR 9023, FP2M, CNRS FR 2036, 200 avenue de la République 92000 Nanterre, France |
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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 Rényi and Tsallis entropies of all orders among all strongly log-concave densities. |
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Keywords: | Majorization Transport Integral inequality Cube slicing strongly log-concave density |
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