A direct proof for the positive definiteness of four point metric spaces

2023/10/31 by Gomi, Kiyonori
#51F99 #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2310.20690

Abstract

We provide a direct and elementary proof for the fact that every four point metric space is positive definite, which was first proved by Meckes based on some embedding theorems of metric spaces. As an outcome of the direct proof, we also provide a condition for the magnitude of a finite metric space to obey the inclusion-exclusion principle with respect to a specific choice of subspaces.

Related

• Positive definite metric spaces
• The observable diameter of metric measure spaces and the existence of points of positive measures
• Metric spaces and positive definite functions
• Analysis on Metric Space Q
• Elementary aspects of the geometry of metric spaces