2021/11/19 by Mario A. Goméz, Gómez, Mario, Facundo Mémoli +1
Engineering · Mathematics · #Advanced Optimization Algorithms Research #FOS: Mathematics #Mathematics and Applications #Metric Geometry (math.MG) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2111.10328
openalex publication_date 2021/11/19 · openalex created_date 2022/11/02 · openalex updated_date 2026/07/28
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.