上曲率有界度量空间中的梯形比较不等式

中文摘要 AI 辅助

在CAT(0)空间的四个点中，平面对称梯形是托勒密不等式和列舍特尼亚克（Reshetnyak）不等式同时取等的构型。我们证明，对于由非递减凸函数（导数为凹函数）索引的、介于上述两个不等式之间的整个插值不等式族，对称梯形仍是极值构型，且该函数类可由这一性质刻画。该结果在性质上强于已知的针对此类函数的四点不等式，且以最优常数形式复现了此前仅对幂函数已知的相关结果。此外，我们将分析扩展至κ>0的CAT(κ)空间，推导了正上曲率界下列舍特尼亚克四边形比较不等式与托勒密不等式的变体，二者均带有最优常数。这两个不等式共同导出了CAT(κ)空间中的梯形比较不等式，其中底的乘积相较于κ=0的情形带有一个额外的最优常数因子。

英文摘要

Among four points of a CAT(0) space, the planar symmetric trapezoids are configurations on which Ptolemy's and Reshetnyak's inequalities are both equalities. We show that the symmetric trapezoids remain extremal for the entire family of inequalities interpolating between the two, indexed by the nondecreasing convex functions with concave derivative, and that this function class is characterized by this property. The result is qualitatively stronger than the known quadruple inequalities for this class of functions, and recovers them with their optimal constants, which were previously known only for power functions. Moreover, we extend the analysis to CAT($κ$) spaces with $κ>0$. We derive variants of Reshetnyak's quadrilateral comparison and of Ptolemy's inequality under positive upper curvature bounds, each with the optimal constant. These two inequalities yield the trapezoid comparison inequality in CAT($κ$) spaces, where the product of the bases carries an additional constant factor compared to the $κ=0$ case. Again the constant is optimal.

发表机构

• Technical University of Munich（慕尼黑工业大学）
• Potsdam Institute for Climate Impact Research（波茨坦气候影响研究所）

机构由 AI 辅助整理，请以论文原文为准。