High dimensional footballs are almost flat

Oxford Mathematician Ben Green on how and why he has been pondering footballs in high dimensions.

"A 3-dimensional football is usually a truncated icosahedron. This solid has the virtue of being pleasingly round, hence its widespread use as a football. It is also symmetric in the sense that there is no way to tell two different vertices apart: more mathematically, there is a group of isometries of $\mathbf{R}^3$ acting transitively on the vertices.

In a recent paper I showed that, perhaps surprisingly, high-dimensional footballs are almost flat. More precisely, a finite transitive subset of the unit sphere in $\mathbf{R}^d$ (the vertices of the football) has width bounded above by a constant times $1/\sqrt{\log d}$: this means that you can rotate the football so that the first coordinates of all the points in it satisfy $|x_1| \leq C/\sqrt{\log d}$.

The bound is sharp, because there does exist a $d$-dimensional football whose width is at least a constant times $1/\sqrt{\log d}$. Its vertices consist of all permutations and all sign combinations of $\frac{1}{\sqrt{H_d}}(\pm 1, \pm \frac{1}{\sqrt{2}}, \dots, \pm \frac{1}{\sqrt{d}})$, where $H_d$ is the harmonic mean $\sum_{i = 1}^d \frac{1}{i}$.

I didn't set out to study high-dimensional footballs for their own sake. The question came up in some work that David Conlon and Yufei Zhao (then both at Oxford) were doing on eigenvalues of quasirandom graphs, a topic in combinatorics.

The proof that high-dimensional footballs are almost flat doesn't use any geometry. Rather, it relies on a bit of representation theory, some inequalities having their origin in work of Selberg on the large sieve in number theory, and most importantly some group theory closely related to work done by Oxford's Michael Collins 15 or so years ago. In particular, it depends on the Classification of Finite Simple Groups."