arXiv 2607.040932026-07-07math.FA

巴拿赫空间中无量纲界的最优性

Optimality of no-dimensional bounds in Banach spaces

Pontifícia Universidade Católica do Rio de Janeiro（天主教里约热内卢 Pontifícia 大学）；Holon Institute of Technology（霍隆理工学院）

Grigory Ivanov, Vladimir Kadets

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AI总结：

讨论巴拿赫空间中组合几何无量纲定理的下界构造，利用Maurey - Pisier定理，证明无量纲赫利、拉东、特弗伯格和选择估计中幂次在最优指数处最优，还给出平凡型空间端点结论及无量纲弱ε-网端点障碍。

摘要：

我们讨论了巴拿赫空间中组合几何的几个无量纲定理的下界构造。共同的机制是Maurey - Pisier定理：巴拿赫空间的上确界Rademacher型迫使有限维\(\ell_p\)结构，并且这些模型空间中的标准坐标配置给出了误差项的下界。对于赫利逼近性质，相关类型是对偶空间的类型。对于彩色拉东、彩色特弗伯格、选择和弱\(\varepsilon\)-网陈述，相关类型是原始空间的类型。我们证明了无量纲赫利、拉东、特弗伯格和选择估计中出现的幂次在最优指数处是最优的。如果达到上确界类型，来自相应类型不等式的已知上界估计具有最佳可能的阶。我们还给出了平凡型空间的端点结论。在这种情况下，赫利、拉东、特弗伯格和选择陈述中的误差项不会趋于零。最后，我们证明了平凡型空间中无量纲弱\(\eps\)-网的端点障碍。对于每个固定的基数界，可以在单位球中找到一个有限集，对于该有限集，在固定的正半径以下不存在该大小的近似弱\(\eps\)-网。证明结合了\(\ell_1^N\)中的单纯形示例、关于克内泽尔图色数的洛瓦兹定理以及平凡型空间中\(\ell_1^N\)的有限表示性。

英文摘要：

We discuss lower-bound constructions for several no-dimensional theorems of combinatorial geometry in Banach spaces. The common mechanism is the Maurey--Pisier theorem: the supremal Rademacher type of a Banach space forces finite-dimensional \(\ell_p\)-structures, and standard-coordinate configurations in these model spaces give lower bounds for the error terms. For the Helly approximation property the relevant type is the type of the dual space. For colorful Radon, colorful Tverberg, selection, and weak \(\varepsilon\)-net statements the relevant type is the type of the original space. We show that the powers appearing in the no-dimensional Helly, Radon, Tverberg, and selection estimates are optimal at the supremal-type exponent. If the supremal type is attained, the known upper estimates coming from the corresponding type inequalities have the best possible order. We also include endpoint statements for spaces of trivial type. In this case the error terms in the Helly, Radon, Tverberg, and selection statements cannot tend to zero. Finally, we prove an endpoint obstruction for no-dimensional weak \(\eps\)-nets in spaces of trivial type. For every fixed cardinality bound, one can find a finite set in the unit ball for which no approximate weak \(\eps\)-net of that size exists below a fixed positive radius. The proof combines the simplex example in \(\ell_1^N\), the Lovász theorem on the chromatic number of Kneser's graph, and finite representability of \(\ell_1^N\) in spaces of trivial type.

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