Local Groups in Delone Sets

Abstract

We prove that in an arbitrary Delone set X in the three-dimensional space, the subset X6 of all points from X at which the local group has no rotation axis of order larger than 6 is also a Delone set. Here, under the local group at pointx ∈ X we mean the symmetry group Sx(2R) of the cluster Cx(2R) of x with radius 2R, where R is the radius of the largest ball free of points of X (according to Delone’s empty sphere theory).