The D4 root system is not universally optimal
Henry Cohn, John H. Conway, Noam D. Elkies, Abhinav Kumar
Abstract
We prove that the D4 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 S3, based on numerical computations suggesting that every 5-design consisting of 24 points in S3 is in a 3-parameter family (which we describe explicitly, based on a construction due to Sali) of deformations of the D4 root system.
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