Zong问题：包含单位球的$2d$个半空间
Zong's problem on $2d$ half-spaces containing the unit ball
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中文总结 AI 辅助
本文证明包含单位球的$2d$个半空间交集必含范数不小于$\nsqrt{d}$的点，等号仅对立方体成立，并给出基于高斯象限概率的证明及扩展。
中文摘要 AI 辅助
我们证明了在$\nmathbb{R}^d$中包含欧几里得单位球的$2d$个半空间的交集包含一个范数至少为$\nsqrt{d}$的点。等号情形刻画了立方体。证明给出了顶点平均平方范数的下界，该下界以其法锥的高斯测度为权重。主要步骤是关于高斯象限概率的一个不等式，通过质量、质心和边界密度的一维比较来证明。我们还讨论了一个二次扩展，并给出了一个独立的谱估计。
英文摘要
We prove that the intersection of $2d$ halfspaces containing the Euclidean unit ball in $\mathbb{R}^d$ contains a point of norm at least $\sqrt{d}$. Equality characterizes the cube. The proof gives a lower bound for the average squared norm of the vertices, weighted by the Gaussian measures of their normal cones. The main step is an inequality for Gaussian orthant probabilities, proved by a one-dimensional comparison of mass, centroid, and boundary density. We also discuss a quadratic extension and give an independent spectral estimate.
