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The D_4 root system is not universally optimal

Henry Cohn·John H. Conway·Noam D. Elkies·Abhinav Kumar
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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.

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Henry Cohn, John H. Conway, Noam D. Elkies, Abhinav Kumar. 2008-09-09. The D_4 root system is not universally optimal. https://doi.org/10.1080/10586458.2007.10129008

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