Abstract
The positive definiteness is a property of a metric space which ensures nice behaviour of its magnitude. For any four-point metric space, the positive definiteness was first established by Meckes, where an embedding theorem is invoked. The aim of my talk is to explain a direct proof for this fact without invoking an embedding theorem. I also discuss a possible condition for the magnitude of a finite metric space to obey the inclusion-exclusion principle with respect to a specific choice of subspaces. This condition is suggested by the direct proof, and its validity is verified when the number of points is small.
Preprint can be found on arxiv.
Introduction
A finite metric is positive deifnite if its similarity matrix is positive definite (as a symmetric matrix).
Ensures nice behaviours.
The direct proof
1, 2, and 3 point metric spaces are positive definite. The proof are quite simple. Every 4 point metric space is positive definite (Meckes). The usual proof uses external mathematics. We show this using a more direct proof.
The inclusion-exclusion principle
Under which circumstances does the magnitude fulfi the property. During the calculation of the direct proof we get a key formula. The fulfillment of this equation is a condition to satisfy the inclusio-exclusion-principle.