Geodesic
Uniqueness theorem for locally antipodal Delaunay sets
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mathematics, mechanics, and mathematical physics. II, Tome 294 (2016), pp. 230-236

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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. 
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     author = {N. P. Dolbilin and A. N. Magazinov},
     title = {Uniqueness theorem for locally antipodal {Delaunay} sets},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {230--236},
     publisher = {mathdoc},
     volume = {294},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2016_294_a12/}
}
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N. P. Dolbilin; A. N. Magazinov. Uniqueness theorem for locally antipodal Delaunay sets. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mathematics, mechanics, and mathematical physics. II, Tome 294 (2016), pp. 230-236. http://geodesic.mathdoc.fr/item/TM_2016_294_a12/