Abstract
In this paper, we show the following result: if C is an n-dimensional 0-symmetric convex compact set, \(f:C\rightarrow [0,\infty )\) is concave, and \(\phi :[0,\infty )\rightarrow [0,\infty )\) is not identically zero, convex, with \(\phi (0)=0\), then
where |C| denotes the volume of C. If \(\phi \) is strictly convex, equality holds if and only if f is affine, C is a generalized symmetric cylinder and f becomes 0 at one of the basis of C. We exploit this inequality to answer a question of Francisco Santos on estimating the volume of a convex set by means of the volume of a central section of it. Second, we also derive a corresponding estimate for log-concave functions.
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Acknowledgements
I would like to thank David Alonso-Gutiérrez, Matthieu Fradelizi, Sasha Litvak, Kasia Wyczesany, Rafael Villa, and Vlad Yaskin for useful remarks and discussions, and Francisco Santos for sharing the question that motivated this article. I would also like to thank the invaluable comments and corrections of the anonymous referee that enormously improved the presentation of this article.
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This research is a result of the activity developed within the framework of the Programme in Support of Excellence Groups of the Región de Murcia, Spain, by Fundación Séneca, Science and Technology Agency of the Región de Murcia. Partially supported by Fundación Séneca project 19901/GERM/15, Spain, and by MICINN Project PGC2018-094215-B-I00 Spain.
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González Merino, B. On a Generalization of the Hermite–Hadamard Inequality and Applications in Convex Geometry. Mediterr. J. Math. 17, 146 (2020). https://doi.org/10.1007/s00009-020-01587-3
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DOI: https://doi.org/10.1007/s00009-020-01587-3