The $D_4$ Root System Is Not Universally Optimal

Abstract

We prove that the $D_4$ root system (equivalently, the set of vertices of the regular 24-cell) is not a universally optimal spherical code. We further conjecture that there is no universally optimal spherical code of 24 points in $S^3$, based on numerical computations suggesting that every 5-design consisting of 24 points in $S^3$ is in a 3-parameter family (which we describe explicitly, based on a construction due to Sali) of deformations of the $D_4$ root system.