On the regular systems

The regular system is the direct generalization of the concept of the integer lattice.
A set of points $X\subset \mathbb{R}^d$ is called Delaunay set if two following conditions hold for some positive $r$ and $R$:
(1) the ball $B_{y}(r)$ of radius $r$ centered at the point $y\in \mathbb{R}^d$ contains at most one point $x\in X$;
(2) the ball $B_{y}(R)$ of radius $R$ contains at least one point $x\in X$.
A lattice of the rank $d$ can be defined as a Delaunay set in $\mathbb{R}^d$ with a point-transitive group of translations.
A regular system is defined as a Delaunay set with a point-transitive group of isometries.
The class of regular systems is of great importance because these sets are considered as the models of the atomic structure of a crystalline matter.
The aim of Local theory [1] is to prove rigorously the existence of a point-transitive group for a Delaunay set $X$ from pairwise congruence of neighborhoods of points of $X$. This problem is related to attempt of explaining why the atomic structure of a matter moves from amorphous state into structure with a rich symmetry group during the phase transition from liquid to solid state [2].
In the talk, we will discuss several key results of the local theory of regular systems [3]-[5].
[1] B.N. Delone, N.P. Dolbilin, M.I. Štogrin, R.V. Galiulin, A local test for the regularity of a system of points. (Russian) Dokl. Akad. Nauk SSSR. 227:1 (1976). P. 19 –- 21.
[2] N.P. Dolbilin, J.C. Lagarias, M. Senechal, Multiregular point systems. Discrete Comput. Geom. 20:4 (1998). P. 477 – 498.
[3] N.P. Dolbilin, Crystal criterion and antipodal Delaunay sets. Vestnik Chelyabinsk. Gos. Univ. 17 (2015). P. 6 – 17.
[4] N.P. Dolbilin, A.N. Magazinov, Uniqueness theorem for locally antipodal Delaunay sets. Proc. Steklov Inst. Math. 294 (2016). P. 215 -– 221.
[5] N. Dolbilin, Delone Sets: Local Identity and Global Order. Volume dedicated to the 60th anniversary of Professors Karoly Bezdek and Egon Schulte, Springer Contributed Volume on Discrete Geometry and Symmetry. Springer, 2016 (to appear). arXiv: 1608.06842