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

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.

arXiv commentLaTex 2e, 17 pages; a section about the inclusion-exclusion principle added