四点条件:托勒密不等式的初等热带化
度量几何
2023-02-07 v3
摘要
托勒密不等式是欧几里得空间中四点间距离的经典关系。六段距离之间的另一关系是四点条件,即连接度量(或加权)树中任意四点的六条路径长度所满足的不等式。四点条件也刻画了有限度量空间何时可嵌入此类树中。细心观察者可能意识到这些不等式形式相似:若将以加法与乘法分别替换为最大值与加法,则由托勒密不等式可得四点条件。我们表明这一相似性不止是巧合。我们识别了CAT空间中由实数参数化的托勒密不等式族,并表明当参数趋于负无穷时涉及这些不等式的某一极限给出四点条件,从而初等地证明后者是托勒密不等式的热带化。
引用
@article{arxiv.2111.10328,
title = {The Four Point Condition: An Elementary Tropicalization of Ptolemy's Inequality},
author = {Mario Gómez and Facundo Mémoli},
journal= {arXiv preprint arXiv:2111.10328},
year = {2023}
}
备注
Improved the introduction by adding the history of Ptolemy's inequality and related works, context for tropical geometry, and Gromov hyperbolicity. Added a dedicated section for definitions. Improved the bounds in Example 2.5 (used to be Example 3.3)