Open Access
2007 The $D_4$ Root System Is Not Universally Optimal
Henry Cohn, John H. Conway, Noam D. Elkies, Abhinav Kumar
Experiment. Math. 16(3): 313-320 (2007).


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. "The $D_4$ Root System Is Not Universally Optimal." Experiment. Math. 16 (3) 313 - 320, 2007.


Published: 2007
First available in Project Euclid: 7 March 2008

zbMATH: 1137.05020
MathSciNet: MR2367321

Primary: 05B40 , 52C17
Secondary: 52A40

Keywords: $24$-cell , $D_4$ root system , potential energy minimization , spherical code , spherical design , universally optimal code

Rights: Copyright © 2007 A K Peters, Ltd.

Vol.16 • No. 3 • 2007
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