Abstract:
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.
Citation:
N. P. Dolbilin, A. N. Magazinov, “Uniqueness theorem for locally antipodal Delaunay sets”, Modern problems of mathematics, mechanics, and mathematical physics. II, Collected papers, Trudy Mat. Inst. Steklova, 294, MAIK Nauka/Interperiodica, Moscow, 2016, 230–236; Proc. Steklov Inst. Math., 294 (2016), 215–221
This publication is cited in the following 7 articles:
M. I. Shtogrin, “On a convex polyhedron in a regular point system”, Izv. Math., 86:3 (2022), 586–619
N. P. Dolbilin, M. I. Shtogrin, “Delone Sets and Tilings: Local Approach”, Proc. Steklov Inst. Math., 318 (2022), 65–89
M. Bouniaev, N. Dolbilin, “The local theory for regular systems in the context of tt -bonded sets”, Symmetry, 10:5 (2018), 159
N. P. Dolbilin, “Delone sets in R3 with 2R-regularity conditions”, Proc. Steklov Inst. Math., 302 (2018), 161–185
I. A. Baburin, M. Bouniaev, N. Dolbilin, N. Yu. Erokhovets, A. Garber, S. V. Krivovichev, E. Schulte, “On the origin of crystallinity: a lower bound for the regularity radius of Delone sets”, Acta Crystallogr. Sect. A, 74:6 (2018), 616–629
N. Dolbilin, “Delone sets with congruent clusters”, Struct. Chem., 27:6 (2016), 1725–1732
N. P. Dolbilin, “Delone sets in R3: regularity conditions”, J. Math. Sci., 248:6 (2020), 743–761
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