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Functional inequalities related to Mahler’s conjecture

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Abstract

Extending a result of Meyer and Reisner (Monatsh Math 125:219–227, 1998), we prove that if \({g: \mathbb{R}\to \mathbb{R}_+}\) is a function which is concave on its support, then for every m > 0 and every \({z\in\mathbb{R}}\) such that g(z) > 0, one has

$$ \int\limits_{\mathbb{R}} g(x)^mdx\int\limits_{\mathbb{R}} (g^{*z}(y))^m dy\ge \frac{(m+2)^{m+2}}{(m+1)^{m+3}},$$

where for \({y\in \mathbb{R}}\) , \({g^{*z}(y)=\inf_x \frac{(1-(x-z)y)_+}{g(x)}}\). It is shown how this inequality is related to a special case of Mahler’s conjecture (or inverse Santaló inequality) for convex bodies. The same ideas are applied to give a new (and simple) proof of the exact estimate of the functional inverse Santaló inequality in dimension 1 given in Fradelizi and Meyer (Adv Math 218:1430–1452, 2008). Namely, if \({\phi:\mathbb{R}\to\mathbb{R}\cup\{+\infty\}}\) is a convex function such that \({0 < \int e^{-\phi} < +\infty}\) then, for every \({z\in\mathbb{R}}\) such that \({\phi(z) < +\infty}\) , one has

$$ \int\limits_{\mathbb{R}}e^{-\phi}\int\limits_{\mathbb{R}} e^{-\mathcal{L}^z\phi}\ge e,$$

where \({\mathcal {L}^z\phi}\) is the Legendre transform of \({\phi}\) with respect to z.

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Correspondence to M. Meyer.

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Fradelizi, M., Meyer, M. Functional inequalities related to Mahler’s conjecture. Monatsh Math 159, 13–25 (2010). https://doi.org/10.1007/s00605-008-0064-0

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  • DOI: https://doi.org/10.1007/s00605-008-0064-0

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