The magnitude of odd balls via Hankel determinants of reverse Bessel polynomials

The magnitude of odd balls via Hankel determinants of reverse Bessel polynomials, Discrete Analysis 2020:5, 42 pp.

This paper concerns an invariant of metric spaces known as magnitude, which, as its name suggests, is a notion of size. It has its origins in category theory and was developed by Tom Leinster, though as it happens the same notion had been developed earlier by Andrew Solow and Stephen Polasky for completely different reasons: they were looking for a natural way to measure biodiversity.

The connection between category theory and metric spaces is via the notion of an enriched category, which is roughly speaking a category where the morphisms between two objects form not just a set but a set with additional structure. If $X$ is a metric space, then we can think of its elements as “objects” and for any two objects $x$ and $y$ we can regard a real number $r$ as a “morphism” if $r\geq d(x,y)$, using addition to compose morphisms. For composition to be possible, we need it to be the case that if $r\geq d(x,y)$ and $s\geq d(y,z)$, then $r+s\geq d(x,z)$, but that requirement is equivalent to the triangle inequality. (This is not how the definitions actually work, but it is a quick way of indicating how there could be a connection between categories and metric spaces.)

It turns out that Euler characteristic can be generalized to finite categories and provides a fundamental notion of size. Pursuing this idea and generalizing it further to enriched categories led Leinster to formulate the following notion of magnitude for a finite metric space. Let $X$ be a finite metric space and define a matrix indexed by $X^2$ by taking its $xy$ entry to be $e^{-d(x,y)}$. Assuming that this matrix is invertible, then the magnitude of $X$ is the sum of the entries of the inverse. (In general, not all finite metric spaces have magnitude, though the definition can be extended to many finite metric spaces for which the above matrix is not invertible, including all finite subsets of Euclidean spaces.)

There are two natural ways to generalize this notion from finite metric spaces to compact metric spaces. One is to consider discretizations, and the other is to replace matrices by kernels and sums by integrals. Mark Meckes showed that the two approaches give essentially the same results, so there is a well-established notion of magnitude for compact metric spaces as well. It appears that magnitude captures many other important metric notions. For instance, if $A$ is a compact subset of $\mathbb R^n$, one can look at the growth rate of the magnitude of $tA$. For many sets $A$, this function contains a great deal of interesting information about $A$ that appears to relate closely to existing well-known invariants, though this is an area that is still being actively investigated.

This paper considers the case of balls in Euclidean space of odd dimension. An algorithm for calculating the magnitude was given by Juan Antonio Barceló and Tony Carbery, who proved that it was a rational function of the radius and calculated the function for a few low dimensions. For instance, they showed that the magnitude of a 5-dimensional ball of radius $r$ is

$$\frac{r^6+18r^5+135r^4+535r^3+1080r^2+1080r+360}{5!(r+3)}.$$

It is difficult to look at such an expression without wondering whether the coefficients can be interpreted somehow – in particular, it is notable that in the examples calculated by Barceló and Carbery they are all positive. This paper derives a formula for the magnitude in general dimension (as opposed to an algorithm for calculating the formula in any given dimension), which allows the author to show that the coefficients are indeed always positive, as well as giving them an interpretation. This is done using Hankel determinants of reverse Bessel polynomials, which are defined on page 3 of the paper, which then open the way to more combinatorial interpretations in terms of objects called weighted Schröder paths.