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The Four Point Condition: An Elementary Tropicalization of Ptolemy's Inequality.
- Source :
-
American Mathematical Monthly . Mar2024, Vol. 131 Issue 3, p187-203. 17p. - Publication Year :
- 2024
-
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. [ABSTRACT FROM AUTHOR]
- Subjects :
- *METRIC spaces
*REAL numbers
*COINCIDENCE theory
*COINCIDENCE
Subjects
Details
- Language :
- English
- ISSN :
- 00029890
- Volume :
- 131
- Issue :
- 3
- Database :
- Academic Search Index
- Journal :
- American Mathematical Monthly
- Publication Type :
- Academic Journal
- Accession number :
- 175670503
- Full Text :
- https://doi.org/10.1080/00029890.2023.2285695