Geometric interpretation of magnitude

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.