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How dense can a finite set of points on the $d$-dimensional unit sphere be if I require that the symmetry group of that arrangement is still transitive on the points?

As a measure for density I use the radius of the largest spherical cap not containing any point in its interior.

For $d=1$ we can get arbitrarily dense. For $d=2$ I suppose the densest set is some orbit of the icosahedral group. Is there something known about general $d$? E.g. is there any $d>1$ for which one can get arbitrarily dense again?

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  • $\begingroup$ The Archimedean polyhedra are the place to start. I'm not actually sure that the convex hull of such a point set has to be Archimedean, but it's certainly plausible that it would. $\endgroup$
    – Steven Stadnicki
    Commented Jun 16, 2020 at 23:48
  • $\begingroup$ @StevenStadnicki I expected something like this. I wonder if there is anything denser than these point sets. $\endgroup$
    – M. Winter
    Commented Jun 17, 2020 at 7:11

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