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M. Winter
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How dense can a transitive sets of points be?

How dense can a 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?

M. Winter
  • 13.6k
  • 3
  • 29
  • 70