基于边界调控的双曲型度量构造及其几何分析OA

This article proposes a boundary-controlled hyperbolic-type metric to address the limitations of classical hyperbolic metric in high-dimensional and general metric spaces.By introducing a boundary dis-tance adjustment factor,a clear definition and construction method for the new metric are provided,and its positive definiteness,symmetry,the triangle inequality,and completeness are proved.This study shows that under this metric,identity mappings exhibit quasi-conformality,and the metric balls in the upper half space and the unit ball are convex.