# The $$D_4$$ Root System is Not Universally Optimal

 Title: The $$D_4$$ Root System is Not Universally Optimal Author: Cohn, Henry; Kumar, Abhinav; Conway, John H.; Elkies, Noam Note: Order does not necessarily reflect citation order of authors. Citation: Cohn, Henry, John H. Conway, Noam D. Elkies, and Abhinav Kumar. 2007. The $$D_4$$ root system is not universally optimal. Experimental Mathematics 16(3): 313-320. Full Text & Related Files: Elkies - The D4 Root System.pdf (171.0Kb; PDF) 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. Published Version: http://akpeters.metapress.com/content/n1700h637u4tk136 Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:2794814

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