On a Blaschke-Santaló-type inequality for $r$-ball bodies
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866914625304395776 |
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| author | Bezdek, Károly |
| author_facet | Bezdek, Károly |
| contents | Let ${\mathbb E}^d$ denote the $d$-dimensional Euclidean space. The $r$-ball body generated by a given set in ${\mathbb E}^d$ is the intersection of balls of radius $r$ centered at the points of the given set. The author [Discrete Optimization 44/1 (2022), Paper No. 100539] proved the following Blaschke-Santaló-type inequality for $r$-ball bodies: for all $0<k< d$ and for any set of given $d$-dimensional volume in ${\mathbb E}^d$ the $k$-th intrinsic volume of the $r$-ball body generated by the set becomes maximal if the set is a ball. In this note we give a new proof showing also the uniqueness of the maximizer. Some applications and related questions are mentioned as well. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_14155 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On a Blaschke-Santaló-type inequality for $r$-ball bodies Bezdek, Károly Metric Geometry Let ${\mathbb E}^d$ denote the $d$-dimensional Euclidean space. The $r$-ball body generated by a given set in ${\mathbb E}^d$ is the intersection of balls of radius $r$ centered at the points of the given set. The author [Discrete Optimization 44/1 (2022), Paper No. 100539] proved the following Blaschke-Santaló-type inequality for $r$-ball bodies: for all $0<k< d$ and for any set of given $d$-dimensional volume in ${\mathbb E}^d$ the $k$-th intrinsic volume of the $r$-ball body generated by the set becomes maximal if the set is a ball. In this note we give a new proof showing also the uniqueness of the maximizer. Some applications and related questions are mentioned as well. |
| title | On a Blaschke-Santaló-type inequality for $r$-ball bodies |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2305.14155 |