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

TL;DR

This paper demonstrates that the 4-point condition, a key inequality in metric tree spaces, can be derived as a tropical limit of Ptolemy's inequality, revealing a deep connection between classical and tropical geometry.

Contribution

It introduces a family of Ptolemaic inequalities in CAT-spaces and shows how the 4-point condition emerges as their tropical limit, providing an elementary proof of this relationship.

Findings

The 4-point condition is the tropicalization of Ptolemy's inequality.

A family of Ptolemaic inequalities in CAT-spaces is identified.

The limit of these inequalities as a parameter approaches negative infinity yields the 4-point condition.

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.