AI 中文总结
本文为青本空间构造拉格朗日型基并给出插值公式，借此刻画强极化不等式极值构型，得到腔恒等式，还利用他人突破证明完全单调函数的广义高斯积不等式。
AI 中文摘要
在本文中，我们为青本空间\(AO(\mathcal A)\)构造了一个拉格朗日型基，该基由超平面排列\(\mathcal A\)的腔自然索引。此构造依赖于奥利克和寺尾的一个维数定理，并给出了\(AO(\mathcal A)\)元素的插值公式。我们用此公式将强极化不等式中的极值构型刻画为由有限考克斯特反射系统产生的那些构型。还进一步表明插值公式产生了一族“腔恒等式”，包括对强极化问题和高斯积不等式早期证明至关重要的恒等式。最后，我们采用维梅特和格里夫斯最近的突破来证明完全单调函数的广义高斯积不等式。
英文摘要
In this paper, we construct a Lagrange-type basis for the Aomoto space $AO(\mathcal A)$, naturally indexed by the chambers of the hyperplane arrangement $\mathcal A$. The construction relies on a dimension theorem of Orlik and Terao and yields an interpolation formula for elements of $AO(\mathcal A)$. We use this formula to characterize the extremal configurations in the strong polarization inequality as those arising from finite Coxeter reflection systems. We further show that the interpolation formula gives rise to a family of \emph{chamber identities}, including identities that were central to our earlier proof of the strong polarization problem and the Gaussian product inequality. Finally, we adapt the recent breakthrough of Ouimet and Greaves to prove a generalized Gaussian Product Inequality for completely monotone functions.
