Weighted Berwald's Inequality
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866918043282571264 |
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| author | Langharst, Dylan Putterman, Eli |
| author_facet | Langharst, Dylan Putterman, Eli |
| contents | The inequality of Berwald is a reverse-Hölder like inequality for the $p$th average, $p\in (-1,\infty),$ of a non-negative, concave function over a convex body in $\mathbb{R}^n.$ We prove Berwald's inequality for averages of functions with respect to measures that have some concavity conditions, e.g. $s$-concave measures, $s\in \mathbb{R}.$ We also obtain equality conditions; in particular, this provides a new proof for the equality conditions of the classical inequality of Berwald. As applications, we generalize a number of classical bounds for the measure of the intersection of a convex body with a half-space and also the concept of radial means bodies and the projection body of a convex body. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_04438 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Weighted Berwald's Inequality Langharst, Dylan Putterman, Eli Metric Geometry Functional Analysis 52A39, 52A41 (Primary), 28A75 (Secondary) The inequality of Berwald is a reverse-Hölder like inequality for the $p$th average, $p\in (-1,\infty),$ of a non-negative, concave function over a convex body in $\mathbb{R}^n.$ We prove Berwald's inequality for averages of functions with respect to measures that have some concavity conditions, e.g. $s$-concave measures, $s\in \mathbb{R}.$ We also obtain equality conditions; in particular, this provides a new proof for the equality conditions of the classical inequality of Berwald. As applications, we generalize a number of classical bounds for the measure of the intersection of a convex body with a half-space and also the concept of radial means bodies and the projection body of a convex body. |
| title | Weighted Berwald's Inequality |
| topic | Metric Geometry Functional Analysis 52A39, 52A41 (Primary), 28A75 (Secondary) |
| url | https://arxiv.org/abs/2210.04438 |