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If by linear or semi-definite programming this number could be shown to be less than 24, then this could be a route to showing that the 24-cell (consisting of 12 antipodal pairs of spheres) is the unique kissing configuration in four dimensions.

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  • $\begingroup$ What do you mean by "placed on the surface". In a touching configuration? $\endgroup$
    – M. Winter
    Commented Nov 9 at 12:06
  • $\begingroup$ Yes. Is the kissing number for mutually non-antipodal unit spheres in four dimensions less than 24? $\endgroup$
    – ELA
    Commented Nov 9 at 22:24

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