On $k$-convex hulls
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916977675599872 |
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| author | Ravasini, Davide |
| author_facet | Ravasini, Davide |
| contents | For every integer $k\geq 2$ and every $R>1$ one can find a dimension $n$ and construct a symmetric convex body $K\subset\mathbb{R}^n$ with $\text{diam}\,Q_{k-1}(K)\geq R\cdot\text{diam}\,Q_k(K)$, where $Q_k(K)$ denotes the $k$-convex hull of $K$. The purpose of this short note is to show that this result due to E.\ Kopecká is impossible to obtain if one additionally requires that all isometric images of $K$ satisfy the same inequality. To this end, we introduce the dual construction to the $k$-convex hull of $K$, which we call the $k$-cross approximation of $K$. We also prove an infinite-dimensional version of the main result that holds in general Hilbert spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_14195 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On $k$-convex hulls Ravasini, Davide Metric Geometry Functional Analysis 52A20, 52A23 For every integer $k\geq 2$ and every $R>1$ one can find a dimension $n$ and construct a symmetric convex body $K\subset\mathbb{R}^n$ with $\text{diam}\,Q_{k-1}(K)\geq R\cdot\text{diam}\,Q_k(K)$, where $Q_k(K)$ denotes the $k$-convex hull of $K$. The purpose of this short note is to show that this result due to E.\ Kopecká is impossible to obtain if one additionally requires that all isometric images of $K$ satisfy the same inequality. To this end, we introduce the dual construction to the $k$-convex hull of $K$, which we call the $k$-cross approximation of $K$. We also prove an infinite-dimensional version of the main result that holds in general Hilbert spaces. |
| title | On $k$-convex hulls |
| topic | Metric Geometry Functional Analysis 52A20, 52A23 |
| url | https://arxiv.org/abs/2411.14195 |