The Four Point Condition: An Elementary Tropicalization of Ptolemy's Inequality

Mario Gómez; Facundo Mémoli

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arXiv · 2111.10328

The Four Point Condition: An Elementary Tropicalization of Ptolemy's Inequality

Mario Gómez·Facundo Mémoli

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Abstract

Ptolemy's inequality is a classic relationship between the distances among four points in Euclidean space. Another relationship between six distances is the 4-point condition, an inequality satisfied by the lengths of the six paths that join any four points of a metric (or weighted) tree. The 4-point condition also characterizes when a finite metric space can be embedded in such a tree. The curious observer might realize that these inequalities have similar forms: if one replaces addition and multiplication in Ptolemy's inequality with maximum and addition, respectively, one obtains the 4-point condition. We show that this similarity is more than a coincidence. We identify a family of Ptolemaic inequalities in CAT-spaces parametrized by a real number and show that a certain limit involving these inequalities, as the parameter goes to negative infinity, yields the 4-point condition, giving an elementary proof that the latter is the tropicalization of Ptolemy's inequality.

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BibTeX RIS

Mario Gómez, Facundo Mémoli. 2023-02-05. The Four Point Condition: An Elementary Tropicalization of Ptolemy's Inequality. https://arxiv.org/abs/2111.10328

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