The D4 root system is not universally optimal
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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