A Comprehensive Definition of the Geometric Mean of Convex Bodies Based on Relations Between Their $p$-Means
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866913337600638976 |
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| author | Brandenberg, René Grundbacher, Florian |
| author_facet | Brandenberg, René Grundbacher, Florian |
| contents | In light of the log-Brunn-Minkowski conjecture, various attempts have been made to define the geometric mean of convex bodies. Many of these constructions are fairly complex and/or fail to satisfy some natural properties one would expect of such a mean. We remedy this by providing a new geometric mean that is both technically simple and inherits all the natural properties expected. To improve our understanding of potential geometric mean definitions, we first study general $p$-means of convex bodies, with the usual definition extended to two series ranging over all $p$ in the extended reals. We characterize their equality cases and obtain (in almost all instances tight) inequalities that quantify how well these means approximate each other. As a corollary, we establish that every Minkowski centered body is equidistant from all its $p$-symmetrizations with respect to the Banach-Mazur distance. Finally, we show that our geometric mean satisfies all the properties considered in recent literature and extend this list with some properties regarding symmetrization and asymmetry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_17512 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Comprehensive Definition of the Geometric Mean of Convex Bodies Based on Relations Between Their $p$-Means Brandenberg, René Grundbacher, Florian Metric Geometry 52A21 (Primary) 52A40 (Secondary) In light of the log-Brunn-Minkowski conjecture, various attempts have been made to define the geometric mean of convex bodies. Many of these constructions are fairly complex and/or fail to satisfy some natural properties one would expect of such a mean. We remedy this by providing a new geometric mean that is both technically simple and inherits all the natural properties expected. To improve our understanding of potential geometric mean definitions, we first study general $p$-means of convex bodies, with the usual definition extended to two series ranging over all $p$ in the extended reals. We characterize their equality cases and obtain (in almost all instances tight) inequalities that quantify how well these means approximate each other. As a corollary, we establish that every Minkowski centered body is equidistant from all its $p$-symmetrizations with respect to the Banach-Mazur distance. Finally, we show that our geometric mean satisfies all the properties considered in recent literature and extend this list with some properties regarding symmetrization and asymmetry. |
| title | A Comprehensive Definition of the Geometric Mean of Convex Bodies Based on Relations Between Their $p$-Means |
| topic | Metric Geometry 52A21 (Primary) 52A40 (Secondary) |
| url | https://arxiv.org/abs/2312.17512 |