发表机构
The University of Texas at Dallas（德克萨斯大学达拉斯分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文将三角形中Steiner内切椭圆的角面积刻画推广到任意维单纯形，证明点位于John椭球边界当且仅当各角单纯形体积的\\(2/n\\)次方之和等于\\(V^{2/n}/n\\)，并联系中心二阶矩。
AI 中文摘要
对于面积为\\(T\\)的三角形，一个平面角面积刻画指出：内点\\(M\\)位于Steiner内切椭圆上当且仅当通过\\(M\\)且平行于各边的直线所截出的三个角三角形的面积\\(T_1,T_2,T_3\\)满足\\(T_1+T_2+T_3=\frac12 T\\)。我们给出任意维数单纯形的相应陈述。若\\(S\\)是体积为\\(V\\)的非退化\\(n\\)-单纯形，且\\(V_1(M),\ldots,V_{n+1}(M)\\)是由\\(M\\)确定的平行于各面的角单纯形的体积，则\\(M\in\partial E_J(S)\\)当且仅当\\(\sum_{i=1}^{n+1}V_i(M)^{2/n}=\frac1n V^{2/n}\\)，其中\\(E_J(S)\\)是\\(S\\)的John椭球。我们还把整个角体积泛函等同于均匀单纯形的中心二阶矩二次型。这里声称的新颖性仅限于角体积公式及其与平面Steiner内切椭圆结果的联系；其下的重心、协方差和John椭球事实是经典的。
英文摘要
For a triangle of area \(T\), a planar corner-area characterization states that an interior point \(M\) lies on the Steiner inellipse precisely when the three corner triangles cut off by the lines through \(M\) parallel to the sides have areas \(T_1,T_2,T_3\) satisfying $$ T_1+T_2+T_3=\frac12 T. $$ We give the corresponding statement for a simplex in arbitrary dimension. If \(S\) is a nondegenerate \(n\)-simplex of volume \(V\) and \(V_1(M),\ldots,V_{n+1}(M)\) are the volumes of the facet-parallel corner simplices determined by \(M\), then $$ M\in\partial E_J(S) \quad\Longleftrightarrow\quad \sum_{i=1}^{n+1}V_i(M)^{2/n}=\frac1n V^{2/n}, $$ where \(E_J(S)\) is the John ellipsoid of \(S\). We also identify the entire corner-volume functional with the central second-moment quadratic of the uniform simplex. The novelty claimed here is limited to the corner-volume formulations and their connections with the planar Steiner-inellipse result; the underlying barycentric, covariance, and John-ellipsoid facts are classical.
