On Banach Spaces with the Helly Approximation Property
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908915473580032 |
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| author | Ivanov, Grigory |
| author_facet | Ivanov, Grigory |
| contents | Qualitatively, a no-dimensional Helly-type theorem says that if every small subfamily of convex sets has a common point in a bounded region, then suitable neighborhoods of all the sets in the whole family have a common point. Quantitative bounds, when available, depend on the ambient metric. We say that a Banach space has the Helly approximation property if the radii of these neighborhoods tend to zero as the size of the subfamilies tends to infinity.
In this paper, we show that the Helly approximation property holds if and only if the dual space has non-trivial Rademacher type. The argument combines Maurey's empirical method with a duality argument at a minimizer of the maximal distance function. We also prove a colorful version of this theorem, with control over the average of the radii. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_22743 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Banach Spaces with the Helly Approximation Property Ivanov, Grigory Functional Analysis Primary 52A35, Secondary 46B07, 46B09, 52A27 Qualitatively, a no-dimensional Helly-type theorem says that if every small subfamily of convex sets has a common point in a bounded region, then suitable neighborhoods of all the sets in the whole family have a common point. Quantitative bounds, when available, depend on the ambient metric. We say that a Banach space has the Helly approximation property if the radii of these neighborhoods tend to zero as the size of the subfamilies tends to infinity. In this paper, we show that the Helly approximation property holds if and only if the dual space has non-trivial Rademacher type. The argument combines Maurey's empirical method with a duality argument at a minimizer of the maximal distance function. We also prove a colorful version of this theorem, with control over the average of the radii. |
| title | On Banach Spaces with the Helly Approximation Property |
| topic | Functional Analysis Primary 52A35, Secondary 46B07, 46B09, 52A27 |
| url | https://arxiv.org/abs/2603.22743 |