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On Grünbaum Type Inequalities

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Abstract

Given a compact set \(K\subset {\mathbb {R}}^n\) of positive volume, and fixing a hyperplane H passing through its centroid, we find a sharp lower bound for the ratio \(\mathrm {vol}(K^{-})/\mathrm {vol}(K)\), depending on the concavity nature of the function that gives the volumes of cross-sections (parallel to H) of K, where \(K^{-}\) denotes the intersection of K with a halfspace bounded by H. When K is convex, this inequality recovers a classical result by Grünbaum. To this respect, we also show that the log-concave case is the limit concavity assumption for such a generalization of Grünbaum’s inequality.

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Acknowledgements

We would like to thank the anonymous referee for her/his very valuable comments and remarks. We also thank Prof. M. A. Hernández Cifre for carefully reading the manuscript and her very helpful suggestions during the preparation of it.

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  1. Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100, Murcia, Spain

    Francisco Marín Sola & Jesús Yepes Nicolás

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  1. Francisco Marín Sola
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  2. Jesús Yepes Nicolás
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Correspondence to Jesús Yepes Nicolás.

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The work is partially supported by MICINN/FEDER project PGC2018-097046-B-I00 and by “Programa de Ayudas a Grupos de Excelencia de la Región de Murcia”, Fundación Séneca, 19901/GERM/15.

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Marín Sola, F., Yepes Nicolás, J. On Grünbaum Type Inequalities. J Geom Anal 31, 9981–9995 (2021). https://doi.org/10.1007/s12220-021-00635-y

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  • DOI: https://doi.org/10.1007/s12220-021-00635-y

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