No-dimensional Helly's theorem in uniformly convex Banach spaces
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866914944423821312 |
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| author | Ivanov, G. |
| author_facet | Ivanov, G. |
| contents | We study the ``no-dimensional'' analogue of Helly's theorem in Banach spaces. Specifically, we obtain the following no-dimensional Helly-type results for uniformly convex Banach spaces: Helly's theorem, fractional Helly's theorem, colorful Helly's theorem, and colorful fractional Helly's theorem.
The combinatorial part of the proofs for these Helly-type results is identical to the Euclidean case as presented in \cite{adiprasito2020theorems}. The primary difference lies in the use of a certain geometric inequality in place of the Pythagorean theorem. This inequality can be explicitly expressed in terms of the modulus of convexity of a Banach space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_05744 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | No-dimensional Helly's theorem in uniformly convex Banach spaces Ivanov, G. Functional Analysis 52A05 (primary), 52A35 We study the ``no-dimensional'' analogue of Helly's theorem in Banach spaces. Specifically, we obtain the following no-dimensional Helly-type results for uniformly convex Banach spaces: Helly's theorem, fractional Helly's theorem, colorful Helly's theorem, and colorful fractional Helly's theorem. The combinatorial part of the proofs for these Helly-type results is identical to the Euclidean case as presented in \cite{adiprasito2020theorems}. The primary difference lies in the use of a certain geometric inequality in place of the Pythagorean theorem. This inequality can be explicitly expressed in terms of the modulus of convexity of a Banach space. |
| title | No-dimensional Helly's theorem in uniformly convex Banach spaces |
| topic | Functional Analysis 52A05 (primary), 52A35 |
| url | https://arxiv.org/abs/2409.05744 |