Uniqueness theorem for locally antipodal Delaunay sets

We prove theorems on locally antipodal Delaunay sets. The main result is the proof of a uniqueness theorem for locally antipodal Delaunay sets with a given $2R$-cluster. This theorem implies, in particular, a new proof of a theorem stating that a locally antipodal Delaunay set all of whose $2R$-clusters are equivalent is a regular system, i.e., a Delaunay set on which a crystallographic group acts transitively.