组合凸性的无维结果。维度再次反击
No-dimensional results of combinatorial convexity. Dimension strikes back
AI总结：
本文讨论了组合凸性中无维版本的Carathéodory和Helly定理，并提出相关开放问题及应用，包括Johnson-Lindenstrauss引理的弱加性类比和量子可行性保证。
摘要：
我们讨论了Carathéodory和Helly定理的无维（近似）版本。我们的目标是引起对这些结果相关开放问题和潜在应用的关注。我们回顾了最近的进展并提出了几个问题。我们还指出了一种简单的方法来『将维度带回画面』：通过将无维陈述与依赖维度的范数比较结合，可以将问题从ℓ₁^d、ℓ∞^d和Schatten类S₁、S∞转移到附近的ℓ^p_d或S_p空间，这些空间具有更好的几何性质。作为基本应用，我们得到了Johnson-Lindenstrauss平坦化引理的弱加性类比，Chebyshev回归在ℓ₁球上的局部到全局估计，以及从局部一致线性测量中获得的量子可行性局部到全局保证。
英文摘要：
We discuss no-dimensional (approximate) versions of Carathéodory's and Helly's theorems. Our goal is to draw attention to open problems and potential applications related to these results. We survey recent progress and pose several questions. We also point out a simple way to ``bring the dimension back into the picture'': by combining no-dimensional statements with dimension-dependent norm comparisons, one can transfer problems in $\ell_1^d$, $\ell_\infty^d$, and Schatten classes $S_1, S_\infty$ to nearby $\ell_p^d$ or $S_p$ spaces with better geometry. As elementary applications, we obtain a weak additive analogue of the Johnson--Lindenstrauss flattening lemma, local-to-global estimates for Chebyshev regression over the $\ell_1$ ball, and a local-to-global guarantee for quantum feasibility from locally consistent linear measurements.
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