单纯形的一个投影恒等式：严格不等式、逆结果与仿射投影
A Projection Identity for Simplices Sharp Inequalities, Converse Results, and Affine Projections
AI总结：
该研究针对欧几里得空间单纯形的投影恒等式，推导了直角单纯形的严格距离不等式，证明了一般单纯形的逆结果，并将相互垂直向量的不等式纳入仿射投影框架，给出了正交性的最小投影子空间数量及最优估计。
AI中文摘要：
我们研究欧几里得空间中单纯形的一个投影恒等式，该恒等式由其单位边方向的框架算子表述。对于直角单纯形，该恒等式导出了一组严格的距离不等式，并完整刻画了等号成立的条件。对于一般单纯形，该公式通过 Ky Fan 原理由 Gram 矩阵的谱控制。我们证明了逆结果，即刻画直角单纯形，并确定了迫使正交性所需的最小投影子空间数量，同时给出了最优定量估计。我们还处理了仿射投影子空间，说明相互垂直向量的原始不等式如何纳入同一框架。
英文摘要：
We study a projection identity for a simplex in Euclidean space, written in terms of the frame operator of its unit edge directions. For a right simplex, the identity leads to a sharp family of distance inequalities and a complete description of equality. For a general simplex, the same formula is controlled by the spectrum of the Gram matrix through the Ky Fan principle. We prove converse results that characterise right simplices and determine the smallest number of projection subspaces needed to force orthogonality, together with an optimal quantitative estimate. We also treat affine projection subspaces and show how the original inequality for mutually perpendicular vectors fits into the same framework.
