On averaged self-distances in finite dimensional Banach spaces
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910707288637440 |
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| author | Lakos, Gyula |
| author_facet | Lakos, Gyula |
| contents | Assume that $\mathfrak A$ is a real Banach space of finite dimension $n\geq2$. Consider any Borel probability measure $ν$ supported on the unit ball $K$ of $\mathfrak A$. We show that \[Δ(ν)=\int_{x \in K}\int_{ y\in K}|x-y|_{\mathfrak A} \,\,\,ν(x)\,ν(y)\leq 2(1-2^{-n}f(n)),\] where $f:\mathbb N\setminus \{0,1\}\rightarrow (0,1]$ is a concrete universal function such that $f(n)\sim \frac{2}{\mathrm e n^2\log n}$. It is hoped that in the estimate`$f(n)$' can be replaced by `$1$'. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_14129 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On averaged self-distances in finite dimensional Banach spaces Lakos, Gyula Functional Analysis Metric Geometry Primary: 52A21, Secondary: 52C17 Assume that $\mathfrak A$ is a real Banach space of finite dimension $n\geq2$. Consider any Borel probability measure $ν$ supported on the unit ball $K$ of $\mathfrak A$. We show that \[Δ(ν)=\int_{x \in K}\int_{ y\in K}|x-y|_{\mathfrak A} \,\,\,ν(x)\,ν(y)\leq 2(1-2^{-n}f(n)),\] where $f:\mathbb N\setminus \{0,1\}\rightarrow (0,1]$ is a concrete universal function such that $f(n)\sim \frac{2}{\mathrm e n^2\log n}$. It is hoped that in the estimate`$f(n)$' can be replaced by `$1$'. |
| title | On averaged self-distances in finite dimensional Banach spaces |
| topic | Functional Analysis Metric Geometry Primary: 52A21, Secondary: 52C17 |
| url | https://arxiv.org/abs/2411.14129 |