三角形Jacobi和民主线性椭圆量的形状（In）相关不等式
Shape (In)dependent Inequalities for Triangleland's Jacobi and Democratic-Linear Ellipticity Quantitities
摘要：
边和中线都是Jacobi坐标量，而且可以均匀地进入Kendall形球上的球坐标和Hopf坐标。这促使在三角形几何和由此产生的形状理论中将中线与边同等对待。因此，在本文中，我们重新制定不等式的中位数的形状量，并着手寻找不等式的质量加权雅可比坐标。这项工作还确定了$4/3$ -权力经常出现在理论的中位数-雅可比群众的比例。其中一个霍普夫坐标是四边形面积。另一个是不等边性，它参数化三角形是左倾还是右倾，并由等边性本身来界定。第三个是椭圆率，它将三角形的高度或平面度参数化为正三角形。而四面积显然是集群选择不变，雅可比坐标，anisoscelesness和椭圆率是集群选择依赖，但可以通过平均所有集群“民主化”。此外，民主化的椭圆率是平凡化的，因为椭圆率是底边和中间二阶矩的差，它们的平均值彼此相等。因此，我们引入了一个独特的“线性椭圆度”量词的高度或平面的三角形，其民主化是非平凡的，并找到不平等的边界。本文的一些不等式是与形状无关的界，而另一些不等式的界依赖于等周比和算术几何边均值比形状变量。
英文摘要：
Sides and medians are both Jacobi coordinate magnitudes, moreover then equably entering the spherical coordinates on Kendall's shape sphere and the Hopf coordinates. This motivates treating medians on the same footing as sides in triangle geometry and the resulting Shape Theory. In this paper, we consequently reformulate inequalities for the medians in terms of shape quantities, and proceed to find inequalities on the mass-weighted Jacobi coordinates. This work moreover identifies the $4/3$ -- powers of which occur frequently in the theory of medians -- as the ratio of Jacobi masses. One of the Hopf coordinates is tetra-area. Another is anisoscelesness, which parametrizes whether triangles are left-or-right leaning as bounded by isoscelesness itself. The third is ellipticity, which parametrizes tallness-or-flatness of triangles as bounded by regular triangles. Whereas tetra-area is clearly cluster choice invariant, Jacobi coordinates, anisoscelesness and ellipticity are cluster choice dependent but can be `democratized' by averaging over all clusters. Democratized ellipticity moreover trivializes, due to ellipticity being the difference of base-side and median second moments, whose averages are equal to each other. Thus we introduce a distinct `linear ellipticity' quantifier of tallness-or-flatness of triangles whose democratization is nontrivial, and find inequalities bounding this. Some of this paper's inequalities are shape-independent bounds, whereas others' bounds depend on the isoperimetric ratio and arithmetic-to-geometric side mean ratio shape variables.
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