A general functional version of Grünbaum's inequality
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866929312257540096 |
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| author | Alonso-Gutiérrez, David Sola, Francisco Marín Goñi, Javier Martín Nicolás, Jesús Yepes |
| author_facet | Alonso-Gutiérrez, David Sola, Francisco Marín Goñi, Javier Martín Nicolás, Jesús Yepes |
| contents | A classical inequality by Grünbaum provides a sharp lower bound for the ratio $\mathrm{vol}(K^{-})/\mathrm{vol}(K)$, where $K^{-}$ denotes the intersection of a convex body with non-empty interior $K\subset\mathbb{R}^n$ with a halfspace bounded by a hyperplane $H$ passing through the centroid $\mathrm{g}(K)$ of $K$.
In this paper we extend this result to the case in which the hyperplane $H$ passes by any of the points lying in a whole uniparametric family of $r$-powered centroids associated to $K$ (depending on a real parameter $r\geq0$), by proving a more general functional result on concave functions.
The latter result further connects (and allows one to recover) various inequalities involving the centroid, such as a classical inequality (due to Minkowski and Radon) that relates the distance of $\mathrm{g}(K)$ to a supporting hyperplane of $K$, or a result for volume sections of convex bodies proven independently by Makai Jr. & Martini and Fradelizi. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_08319 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A general functional version of Grünbaum's inequality Alonso-Gutiérrez, David Sola, Francisco Marín Goñi, Javier Martín Nicolás, Jesús Yepes Metric Geometry Functional Analysis A classical inequality by Grünbaum provides a sharp lower bound for the ratio $\mathrm{vol}(K^{-})/\mathrm{vol}(K)$, where $K^{-}$ denotes the intersection of a convex body with non-empty interior $K\subset\mathbb{R}^n$ with a halfspace bounded by a hyperplane $H$ passing through the centroid $\mathrm{g}(K)$ of $K$. In this paper we extend this result to the case in which the hyperplane $H$ passes by any of the points lying in a whole uniparametric family of $r$-powered centroids associated to $K$ (depending on a real parameter $r\geq0$), by proving a more general functional result on concave functions. The latter result further connects (and allows one to recover) various inequalities involving the centroid, such as a classical inequality (due to Minkowski and Radon) that relates the distance of $\mathrm{g}(K)$ to a supporting hyperplane of $K$, or a result for volume sections of convex bodies proven independently by Makai Jr. & Martini and Fradelizi. |
| title | A general functional version of Grünbaum's inequality |
| topic | Metric Geometry Functional Analysis |
| url | https://arxiv.org/abs/2404.08319 |