关于凸体的随机直径
On random diameters of convex bodies
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中文总结 AI 辅助
本文研究凸体随机截面直径的上下界，并应用于p-椭球体，解决了信息复杂性理论中关于随机信息替代最优信息的自然猜想，给出最优二分法。
中文摘要 AI 辅助
设 $K \subset \mathbb{R}^N$ 为一个包含原点在其内部的凸体。本文研究 $K$ 的随机截面的直径，并利用 $K$ 的几何参数推导出这些直径的上界和下界。我们的界以大概率成立，并为这一广泛研究的课题提供了新的见解。我们的上界补充了所谓的低 $M^*$ 估计，并且在许多情况下它要尖锐得多。我们给出的两个下界各自性质不同：根据所讨论的凸体，每个下界都可能更优，并且在许多有趣的情形下，它们也与上界相匹配。随后，我们将结果应用于确定 $p$-椭球体（$\ell_p$ 球在对角算子下的像）的随机直径，改进了先前已知的结果，并在许多情形下达到了尖锐的估计。一个值得注意的应用是信息复杂性理论，我们针对 Hinrichs、Prochno 和 Sonnleitner 在 2023 年提出的一个非常自然的猜想，成功建立了一个简单（且本质上最优）的二分法。我们的解答精确地确定了何时用随机（高斯）信息替换用于从 $p$-椭球体中恢复向量的最优信息是有用的，而随机信息在实际中可能更容易获取。
英文摘要
Let $K \subset \mathbb{R}^N$ be a convex body containing the origin in its interior. In this work, we study the diameters of random sections of $K$ and derive upper and lower bounds for them in terms of geometric parameters of $K$. Our bounds hold with large probability, and they offer new insights into this widely studied subject. Our upper bound complements the so-called low $M^*$-estimate and in many cases it is much sharper. The two lower bounds that we give are each of a different nature: depending on the body in question each time, either could be better, and in many interesting cases it matches the upper bound too. Subsequently, we apply our results to determine random diameters of $p$-ellipsoids (images of $\ell_p$ balls under diagonal operators), improving upon previously known results and achieving sharp estimates in many cases. One notable application is to Information-Based Complexity Theory, where we manage to establish a simple (and essentially optimal) dichotomy in response to a very natural conjecture posed by Hinrichs, Prochno and Sonnleitner in 2023. Our solution settles precisely when it is useful to replace the optimal information used for the recovery of vectors from a $p$-ellipsoid with random (Gaussian) information, which can be more practical to obtain.
