The $D_4$ Root System Is Not Universally Optimal
Cohn, Henry ; Conway, John H. ; Elkies, Noam D. ; Kumar, Abhinav
Experiment. Math., Tome 16 (2007) no. 1, p. 313-320 / Harvested from Project Euclid
Text on Project Euclid
PDF
Alt PDF
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.
Publié le : 2007-05-15
Classification:  $24$-cell,  $D_4$ root system,  potential energy minimization,  spherical code,  spherical design,  universally optimal code,  52C17,  05B40,  52A40 
@article{1204928532,
     author = {Cohn, Henry and Conway, John H. and Elkies, Noam D. and Kumar, Abhinav},
     title = {The $D\_4$ Root System Is Not Universally Optimal},
     journal = {Experiment. Math.},
     volume = {16},
     number = {1},
     year = {2007},
     pages = { 313-320},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1204928532}
}

Cohn, Henry; Conway, John H.; Elkies, Noam D.; Kumar, Abhinav. The $D_4$ Root System Is Not Universally Optimal. Experiment. Math., Tome 16 (2007) no. 1, pp.  313-320. http://gdmltest.u-ga.fr/item/1204928532/