Seems like the crucial step here was being able to develop a systematic way to address every possible type of 'repulsive force' in a given dimension, which then allowed them to show that the given solution (for example E8 in dimension 8) optimally solved the packing problem for each possible force.<p>They talked about a few examples of the different repulsive forces that were characterised, but I wonder what they mean by <i>all</i> of those forces. It seems like there are an infinite way to parameterise a force between two points given a measure (by taking different functions of the distance between the points) but also an infinite way of choosing a measure.<p>In the article they say<p>> <i>Viazovska, Cohn and their collaborators restricted their attention to the universe of repulsive forces. More specifically, they considered ones that are completely monotonic, meaning (among other things) that the repulsion is stronger when points are closer to each other. This broad family includes many of the forces most common in the physical world. It includes inverse power laws — such as Coulomb’s inverse square law for electrically charged particles — and Gaussians, the bell curves that capture the behavior of entities with many essentially independent repelling parts, such as long polymers. The sphere-packing problem sits at the outer edge of this universe: The requirement that the spheres not overlap translates into an infinitely strong repulsion when their center points are closer together than the diameter of the spheres.</i><p>So I guess that means that it doesn't really matter which kind of measure you use as long as the repulsive force is monotonic. The auxiliary functions they discuss probably incorporate however the measure and force are defined in a nice way, so that the method for constructing the optimal auxiliary function is agnostic towards them.<p>Reading more, in the linked paper's abstract [0], it seems they are only dealing with euclidean distance, and the auxiliary function is points on a sphere:<p>> <i>We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points). Call a configuration sharp if there are m distances between distinct points in it and it is a spherical (2m-1)-design. We prove that every sharp configuration minimizes potential energy for all completely monotonic potential functions. Examples include the minimal vectors of the E_8 and Leech lattices. We also prove the same result for the vertices of the 600-cell, which do not form a sharp configuration. For most known cases, we prove that they are the unique global minima for energy, as long as the potential function is strictly completely monotonic. For certain potential functions, some of these configurations were previously analyzed by Yudin, Kolushov, and Andreev; we build on their techniques. We also generalize our results to other compact two-point homogeneous spaces, and we conclude with an extension to Euclidean space.</i><p>[0] <a href="https://arxiv.org/abs/math/0607446" rel="nofollow">https://arxiv.org/abs/math/0607446</a>