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 establish the following two results for an $n$-point metric space $X$ of negative type : $\lim_{t\to 0}{\rm Mag}\ Z_{tX} = 1$ and ${\rm Mag}\ Z_{X}< n$ for $n>1$. The second result gives a negative answer to the problem given by Gomi--Meckes. 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.