Geometric interpretation of magnitude

TL;DR

This paper provides a geometric interpretation of the magnitude of certain matrices and metric spaces, linking it to the radius of circumspheres and circum-quasi-spheres, and addresses a specific open problem.

Contribution

It introduces a geometric framework for understanding magnitude via circumsphere radii, including for indefinite inner product spaces, and resolves a known problem about magnitude bounds.

Findings

Magnitude of positive definite matrices relates to circumsphere radius.

For n-point metric spaces of negative type, magnitude is less than n.

Provides geometric description for magnitude in indefinite inner product spaces.

Abstract

For an $n\times n$ positive definite symmetric matrix $Z$ with $Z_{ii} = 1$ for all $i$, we show that there exists a set of vectors $V_Z\subset \mathbb{R}^n$ such that the radius $R$ of the circumsphere of $V_Z$ satisfies ${\rm Mag}\ Z = (1-R^2)^{-1}$. This leads us to interpret geometrically several known and new facts on magnitude. In particular, we show that ${\rm Mag}\ Z_{X}< n$ for an $n$-point metric space $X$ of negative type with $n>1$. This result gives a negative answer to a problem given by Gomi--Meckes \cite{GM}. Furthermore, we also have a similar geometric description of magnitude for general real symmetric matrix $Z$ with $Z_{ii} = 1$ for all $i$. In this case, the radius corresponds to that of a circum-quasi-sphere, namely the set of points having a prescribed norm in a vector space endowed with an indefinite inner product.