[Paper Review] On the magnitude of odd balls via potential functions

Simon Willerton|arXiv (Cornell University)|Apr 6, 2018

Advanced Combinatorial Mathematics6 references3 citations

TL;DR

This paper derives an explicit determinantal formula for the magnitude of odd-dimensional balls using potential functions and reverse Bessel polynomials, providing the conceptual pathway that led to the later proof of a Hankel determinant formula. It conjectures a closed-form expression for the derivative of the magnitude in terms of squared Hankel determinants of reverse Bessel polynomials, verified numerically up to dimension 57.

ABSTRACT

Magnitude is a measure of size defined for certain classes of metric spaces; it arose from ideas in category theory. In particular, magnitude is defined for compact subsets of Euclidean space and, in arXiv:1507.02502, Barceló and Carbery gave a procedure for calculating the magnitude of balls in odd dimensional Euclidean spaces. In this paper their approach is modified in various ways: this leads to an explicit determinantal formula for the magnitude of odd balls and leads to the conjecturing of a simpler formula in terms of Hankel determinants. This latter formula is proved using a rather different approach in arXiv:1708.03227, but the current paper provides the reasoning that lead to the formula being conjectured. Finally, an empirically-tested Hankel determinant formula for the derivative of the magnitude is conjectured.

Motivation & Objective

• To explain the heuristic reasoning behind the Hankel determinant formula for the magnitude of odd-dimensional balls, which was previously stated without derivation.
• To provide an alternative approach to computing the magnitude of odd balls using potential functions and recursive differential equations.
• To bridge the gap between Barceló and Carbery's algorithmic method and Willerton's later Hankel determinant formula by revealing the underlying structure.
• To conjecture a new formula for the derivative of the magnitude of odd balls in terms of Hankel determinants, supported by numerical evidence up to dimension 57.

Proposed method

• The paper defines a sequence of functions $\psi_i(r)$ recursively via $\psi_0(r) = e^{-r}$ and $\psi_{i+1}(r) = -\frac{1}{r}\psi_i'(r)$, which generate potential functions for the ball.
• It introduces the reverse Bessel polynomials $\chi_i(R) = e^R R^{2i} \psi_i(R)$, which are polynomials with positive integer coefficients and appear in the determinant formulas.
• Using Cramer’s Rule on a linear system derived from the boundary conditions of the potential function, the paper derives an expression for the first non-trivial derivative of the potential function at the radius $R$.
• The method connects the magnitude of the ball to the determinant of a matrix whose entries are $\chi_{i+j}(R)$, leading to a determinantal expression for the magnitude.
• The paper analyzes the behavior of derivatives of the potential function at the boundary and observes a recurring pattern in the denominators and numerators across dimensions.
• It formulates a conjecture linking the derivative of the magnitude to the square of the Hankel determinant of $\chi_{i+j+1}(R)$, with a normalization factor involving $R^{n-1}/(n-1)!$.

Experimental results

Research questions

• RQ1 How can the magnitude of an odd-dimensional ball be expressed in terms of determinants of a sequence of polynomials arising from potential functions?
• RQ2 What structural pattern underlies the appearance of Hankel determinants in the magnitude formula for odd balls?
• RQ3 Why do the denominators of the potential function's derivatives in dimension $n$ resemble the numerators in dimension $n+2$?
• RQ4 Can the derivative of the magnitude function be expressed as a rational function involving Hankel determinants of reverse Bessel polynomials?
• RQ5 What is the conceptual and computational pathway that leads from Barceló and Carbery’s algorithm to the Hankel determinant formula?

Key findings

• The magnitude of the $n$-ball with $n=2p+1$ is given by a determinantal formula involving the ratio of Hankel determinants of reverse Bessel polynomials $\chi_{i+j}(R)$.
• The first non-trivial derivative of the potential function at $r=R$ is expressed as $-\frac{\det[\chi_{i+j+1}(R)]_{i,j=0}^{p}}{R^{p+1}\det[\chi_{i+j}(R)]_{i,j=0}^{p}}$, derived via Cramer’s Rule.
• The derivative of the magnitude $\frac{d}{dR}|B_R^n|$ is conjectured to equal $\frac{\left(\det[\chi_{i+j+1}(R)]_{i,j=0}^{p}\right)^2}{(2p)!\, R^2\left(\det[\chi_{i+j}(R)]_{i,j=0}^{p}\right)^2}$, verified numerically up to $n=57$.
• The conjectured derivative formula matches the square of the first non-trivial derivative of the potential function scaled by $\frac{R^{n-1}}{(n-1)!}$, suggesting a deep structural link.
• The reverse Bessel polynomials $\chi_i(R)$ exhibit a combinatorial structure with positive integer coefficients, and their determinants encode the magnitude and its derivative.
• The paper identifies a recurring pattern: the denominator of the $p$-th derivative in dimension $n$ matches the numerator of the $(p+1)$-th derivative in dimension $n+2$, though no theoretical explanation is provided.